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Using LINQ to group a list of objects

var groupedCustomerList = CustomerList.GroupBy(u => u.GroupID)
                                      .Select(grp =>new { GroupID =grp.Key, CustomerList = grp.ToList()})
                                      .ToList();

var groupedCustomerList = CustomerList
                         .GroupBy(u => u.GroupID, u=>{
                                                        u.Name = "User" + u.Name;
                                                        return u;
                                                     }, (key,g)=>g.ToList())
                         .ToList();

If you don't want to change the original data, you should add some method (kind of clone and modify) to your class like this:

public class Customer {
  public int ID { get; set; }
  public string Name { get; set; }
  public int GroupID { get; set; }
  public Customer CloneWithNamePrepend(string prepend){
    return new Customer(){
          ID = this.ID,
          Name = prepend + this.Name,
          GroupID = this.GroupID
     };
  }
}    
//Then
var groupedCustomerList = CustomerList
                         .GroupBy(u => u.GroupID, u=>u.CloneWithNamePrepend("User"), (key,g)=>g.ToList())
                         .ToList();

I think you may want to display the Customer differently without modifying the original data. If so you should design your class Customer differently, like this:

public class Customer {
  public int ID { get; set; }
  public string Name { get; set; }
  public int GroupID { get; set; }
  public string Prefix {get;set;}
  public string FullName {
    get { return Prefix + Name;}
  }            
}
//then to display the fullname, just get the customer.FullName; 
//You can also try adding some override of ToString() to your class

var groupedCustomerList = CustomerList
                         .GroupBy(u => {u.Prefix="User", return u.GroupID;} , (key,g)=>g.ToList())
                         .ToList();

is this what you want?

var grouped = CustomerList.GroupBy(m => m.GroupID).Select((n) => new { GroupId = n.Key, Items = n.ToList() });

var result = from cx in CustomerList
         group cx by cx.GroupID into cxGroup
     orderby cxGroup.Key
     select cxGroup; 

foreach (var cxGroup in result) { 
Console.WriteLine(String.Format("GroupID = {0}", cxGroup.Key)); 
  foreach (var cx in cxGroup) { 
    Console.WriteLine(String.Format("\tUserID = {0}, UserName = {1}, GroupID = {2}", 
      new object[] { cx.ID, cx.Name, cx.GroupID })); 
  }
}