Get all Active Directory groups with Manager Can Updated Membership List checkbox.

Get-ManagerCanUpdatedMembershipList

$groups = Get-ADGroup -Filter *
$result = @()

foreach ( $group in $groups ) {
 $object = Get-ADObject `
     -Identity $group.DistinguishedName `
     -Properties managedBy

  if ( $object.managedBy -ne $null ) {
    $managedBy = Get-ADUser -Filter "DistinguishedName -eq '$($object.managedBy)'"
 
    $acl = Get-Acl "AD:\$($object.DistinguishedName)"

    foreach ( $acc in $acl.Access ) {
      $value = $acc.IdentityReference.Value
      $ActiveDirectoryRights = $acc.ActiveDirectoryRights      
      if ($value -like "*\$($managedBy.SamAccountName)") {
        if ($ActiveDirectoryRights -like "WriteProperty") {          
          $result += [PSCUSTOMOBJECT]@{
            GroupName = $object.Name
            managedBy = $managedBy.Name
          }
        }
      }
    }
  }
}

$result | ft -a