The script checks the certificate drive, looking for a certificate with the subject equal to the current machine's name and domain, and then chooses the valid certificate with the latest expiration date.  It then takes that certificate's thumbprint, splits it into an array of byte-sized hex strings, and either updates or creates the relevant Hyper-V registry key with the byte data.

PowerShell
Edit|Remove
# Get the thumbprint from the most recent local machine certificate
$hexstr = @(ls cert:\LocalMachine\Root | `
    ?{$_.subject -eq "CN=$env:computername.$((gwmi win32_computersystem).domain)"} | `
    ?{$_.verify()} | sort -property notafter)[-1].thumbprint
if(!$hexstr){
    throw "Error: Local machine certificate not found."
}
# Convert the thumbprint string into an array of hex byte strings
$hexstrarray = @()
for($i = 0;$i -lt $hexstr.length;$i += 2){
    $hexstrarray += "0x" + $hexstr.substring($i,2)
}
# Check for the existence of the key, updating or creating it as needed
if((gp 'HKLM:\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Virtualization').authcertificatehash){
    set-itemproperty 'HKLM:\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Virtualization' `
        -name AuthCertificateHash -value ([byte[]]$hexstrarray)
}else{
    [void](new-itemproperty 'HKLM:\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Virtualization' `
        -name AuthCertificateHash -value ([byte[]]$hexstrarray))
}