Program.cs

// add reference to System.DirectoryServices.AccountManagement.dll
var sidAttr = pool.Attributes["applicationPoolSid"];
var securityIdentifier = new SecurityIdentifier((string)sidAttr.Value);
using(var ctx = new PrincipalContext(ContextType.Machine))
using(var group = GroupPrincipal.FindByIdentity(ctx, IdentityType.Sid, (string)securityIdentifier.Value))
{
    Debug.Assert(group != null);
}