You need to use a comparison function, they are functions that from two instances of your type return an integer that return 0 if both are equals, a negative value if the first is less than the second and a positive value if the first is greater than the second.

MSDN has a nice table that is easier to follow than text (StackOverflow still doesn’t support tables in 2014)

IComparer<T>

Most sort methods accept a custom comparer implementation of type IComparer<T> you should create one encapsulating your custom rules for Group :

class GroupComparer : IComparer<Group>
{
    public int Compare(Group a, Group b)
    {
        if (a != null && b != null && (a.Id == 0 || b.Id == 0))
        {
            if (a.Id == b.Id)
            {
                // Mandatory as some sort algorithms require Compare(a, b) and Compare(b, a) to be consistent
                return 0; 
            }
            return a.Id == 0 ? -1 : 1;
        }

        if (a == null || b == null)
        {
            if (ReferenceEquals(a, b))
            {
                return 0;
            }
            return a == null ? -1 : 1;
        }
        return Comparer<string>.Default.Compare(a.Name, b.Name);
    }
}

Usage:

items.OrderBy(_ => _, new GroupAuthorityComparer());

IComparable<T>

If it is the only way to compare Group instances you should make it implement IComparable<T> so that no aditional code is needed if anyone want to sort your class :

class Group : IComparable<Group>
{
    ...
    
    public int CompareTo(Group b)
    {
        if (b != null && (Id == 0 || b.Id == 0))
        {
            if (Id == b.Id)
            {
                // Mandatory as some sort algorithms require Compare(a, b) and Compare(b, a) to be consistent
                return 0; 
            }
            return Id == 0 ? -1 : 1;
        }
        
        return Comparer<string>.Default.Compare(Name, b.Name);
    }   
}

Usage:

items.OrderBy(_ => _.Group);

The choice between one way or the other should be done depending on where this specific comparer is used: Is it the main ordering for this type of item or just the ordering that should be used in one specific case, for example only in some administrative view.

You can even go one level up and provide an IComparable<GroupAuthority> implementation (It’s easy once Group implement IComparable<Group>):

class GroupAuthority : IComparable<GroupAuthority>
{
    ...
    
    public int CompareTo(GroupAuthority b)
    {
        return Comparer<Group>.Default.Compare(Group, b.Group);
    }
}

Usage:

items.OrderBy(_ => _);

The advantage of the last one is that it will be used automatically, so code like: GroupAuthoritys.ToList().Sort() will do the correct thing out of the box.