There are many ways of doing this. Here is one way to bring everything into a single consistent behavior, a landing place from which you can vary slightly at need.

This presumes that you have Azure/AD sync installed and working in general, and yielding errors in the Synchronization Service window for one or more users. Make sure that you don’t have duplicate email addresses in AD, that could be bad.

The first steps are in Active Directory Users and Computers.

1. Set the user’s email correctly in his/her AD object, in “E-mail” under General.
2. Set proxyAddresses in the Attribute Editor. The primary email address has to be the same, and in proxyAddresses has to be of the format “SMTP:email@domain.com”. There can be others in proxyAddresses but smtp must be lowercase. Also in proxyAddresses, set mailNickname blank.
3. Under Account, either the user login name plus the dropdown domain is to be the same as the above, or it is to be a valid login according to the O365 console. If the dropdown domain list is local only, you can add the Internet domain list in Active Directory Domains and Trusts, with a right-click on the root level in the left pane of that window, and then an add of one or more alternative UPN suffixes. Then restart ADUC and the domain(s) you just added will be available in the dropdown.

Now we do some other things.

4. Run Azure/AD sync, this is CMD, do it as administrator:

repadmin /syncall /AdeP
@"%SystemRoot%\System32\WindowsPowerShell\v1.0\powershell.exe" -NoProfile -InputFormat None -ExecutionPolicy Bypass -Command ^
"Import-Module ADSync; Start-ADSyncSyncCycle -PolicyType Delta"

5. If you see errors in the Synchronization Service window, you’ll need something called a “hard match”, which changes enough to connect the AD user with the Azure user. Here’s a script for it, in Powershell. You’ll need to run it as administrator on a domain controller, and it will ask you to log into the O365 tenant. Items of note:

• This uses something called UPN, UserPrincipalName. While Azure’s UPN is the O365 “primary email address”, Active Directory’s UPN is the double item under Profile in AD, the username plus the domain dropdown.
• If you use this approach, the Office 365 login is changed to be the same as the AD login — which is a good thing in the net result, it keeps things very consistent and predictable.
• Once the hard match is complete, you can change anything on the AD side and it will mirror up to Azure.

############################################
# Active Directory / Office 365 Hard Match #
############################################

$ADUPN = 'active_directory_user@companynetwork.com'
$AzureUPN = 'azure_user@companynetwork.com'

'############################################'
'# Active Directory / Office 365 Hard Match #'
'############################################'
''

'Initiating prep for hard match.'
"Active Directory : $ADUPN"
"Azure AD :         $AzureUPN"
''

# Sets TLS version.  Necessary for some platforms.
[Net.ServicePointManager]::SecurityProtocol = [Net.SecurityProtocolType]::Tls12

$reportStatus = ''
$currentOp = ''
function ShowProgress {
	param( [string]$reportStatus, [string]$currentOp )

	Write-Progress -Activity "Hard Match" -Status $reportStatus -PercentComplete -1 -CurrentOperation $currentOp
	# Write-Progress is not compatible with some remote shell methods.

}

Function PrepareModule {
	param( [string]$ModuleName )

	If (Get-Module -ListAvailable -Name $ModuleName)
		{ Update-Module $ModuleName }
	Else
		{ Install-Module $ModuleName }
	}

Set-ExecutionPolicy -Scope Process -ExecutionPolicy Unrestricted -Force > $null

'Preparing Powershell environment...'

ShowProgress("Preparing Powershell environment...","Setting up to use Powershell Gallery...")

ShowProgress("Preparing Powershell environment:","Setting up to use page provider NuGet...")
Install-PackageProvider -Name NuGet -Force | Out-Null

# This appears to set PSGallery nicely when need be
Register-PSRepository -Default -InstallationPolicy Trusted 2> $null
Set-PSRepository -InstallationPolicy Trusted -Name PSGallery

ShowProgress("Preparing Powershell environment...","Checking/preparing module NuGet...")
PrepareModule("NuGet")
ShowProgress("Preparing Powershell environment...","Checking/preparing module AzureAD...")
PrepareModule("AzureAD")

''
'Setting up hard match...'
''

'Connect to AzureAD:'
Connect-AzureAD

''
'Turn off AZ/AD Sync...'
''

Set-ADSyncScheduler -SyncCycleEnabled $false

"Now get original Azure ImmutableID for $AzureUPN ..."
$AzureUser = Get-AzureADUser -SearchString $AzureUPN
$OriginalAzureImmutableID = $AzureUser.ImmutableID
"Extracted Azure ImmutableID: $OriginalAzureImmutableID"
""
""
"And now extract AD GUID for $ADUPN ..."
ldifde -f export.txt -r "(Userprincipalname=$ADUPN)" -l *
$ADGUID = (-split (type export.txt | select-string "ObjectGUID"))[1]

''
"Extracted AD GUID: $ADGUID"
""
""
'Set AD GUID as Azure ImmutableID...'
Set-AzureADuser -ObjectID $AzureUser.ObjectID -ImmutableID $ADGUID

''
'New Azure ImmutableID retrieved as confirmation:'
$AzureUser = Get-AzureADUser -SearchString $AzureUPN
$AzureUser.ImmutableID

''
'Finally, turn on AZ/AD Sync again...'

Set-ADSyncScheduler -SyncCycleEnabled $true

'Done!'