Four your information, it is not possible to add an active directory group as owner of SharePoint group. Only an user or existing SharePoint group could.

Here, we will add an user acccount based on the claim “User Principle Name” (UPN)

$claim = New-SPClaimsPrincipalClaimValue “<name.surname@domain.com>” ClaimType http://schemas.xmlsoap.org/ws/2005/05/identity/claims/upnTrustedIdentityTokenIssuer “Trusted_Identity_Provider_Name”

$claimuser = $claim.ToEncodedString()

$web = Get-SPWeb “<url_collection_site>”

$groupname = “<group_name>” 

$groupdescription = “<group_description>

$grouppermission = “<type_of_permission>”   #READ, CONTRIBUTE, FULL CONTROL

$groupowner = $web.SiteUsers[$claimuser“]

$NewSPGroup = $web.SiteGroups.Add($groupname, $groupowner,$web.CurrentUser, $description)

function AddGroupToSite ($web, $groupName, $permLevel)

{

   $account = $web.SiteGroups[$groupName]

   $assignment = New-Object Microsoft.SharePoint.SPRoleAssignment($account)

   $role = $web.RoleDefinitions[$permLevel]

   $assignment.RoleDefinitionBindings.Add($role);

   $web.RoleAssignments.Add($assignment)

}

AddGroupToSite -web $web –groupName $groupnamepermLevel $grouppermission