Cannot add an IIS Application Pool to a local user group via C#
===

I am trying to add an IIS Application Pool user to a local user group remotely via C# and am having some difficulty.

I have tried the two approaches below:

    // This results in a ArgumentNullException because user is never set
    using (PrincipalContext pc = new PrincipalContext(ContextType.Machine, serverName))
    {
        UserPrincipal user = UserPrincipal.FindByIdentity(pc, String.Format(@"IIS APPPOOL\{0}", rootApplicationPoolName));
        GroupPrincipal myGroup = GroupPrincipal.FindByIdentity(pc, "mygroupname");
        myGroup.Members.Add(user);
        myGroup.Save();
    }

Also:

    // This results in a NoMatchingPrincipalException saying the user could not be found
    using (PrincipalContext pc = new PrincipalContext(ContextType.Machine, serverName))
    {
        GroupPrincipal myGroup= GroupPrincipal.FindByIdentity(pc, "mygroupname");
        myGroup.Members.Add(pc, IdentityType.Name, String.Format(@"IIS APPPOOL\{0}", appPoolName));
        myGroup.Save();
    }

I can manually add this user to the group and it works just fine.

What am I missing?