Use SQL to clone a tree structure represented in a database

Assming that ID is an identity column, with auto assigned values. I'm doing this off the cuff, so appologies for any syntax errors. I hope the comments make the intent clear.

/* Find all ancestors for a given ID */
WITH Ancestors(ChildID, AncestorID) AS
(
  SELECT id AS ChildID, id As AncestorID
     FROM YourTable WHERE ParentGroupID=--1
  UNION ALL
  SELECT a.ChildID, d.ParentGroupID
     FROM  Ancestors AS a INNER JOIN
       YourTable d ON d.ID = a.AncestorID
     WHERE (b1.ParentGroupID <> -1))
),
/* Insert new rows for all ancestors of a given ID and save the results, so we have access to the new ID. we also have a column for the old ID. */
IDMap(ID, ParentGroupID, SomeValue, OldID) AS
{
    // insert, using identity column assigned ID, and save the old ID
   INSERT INTO YourTable SELECT d.ParentGroupID, d.SomeValue+"-cloned", d.ID FROM YourTable d
      INNER JOIN Ancestors a ON a.ChildID = d.ID 
      WHERE a.AncestorID=<the ID to clone>
}
/* Now update the parentID in the inserted data to the new ID */
UPDATE YourTable
SET ParentGroupID = map.ID
FROM YouTable t INNER JOIN (SELECT * FROM IDMap) map
  ON t.ParentGroupID=map.OldID

Try this, based on a query from Quassnoi's article Adjacency List vs Nested Sets: SQL Server:

WITH q AS
(
    SELECT  h.*, 1 AS level
    FROM    Table1 h
    WHERE   id = 3
    UNION ALL
    SELECT  hp.*, level + 1
    FROM    q
    JOIN    Table1 hp
    ON      hp.id = q.ParentGroupID
), q2 AS (
    SELECT
        ID,
        ParentGroupID,
        SomeValue,
        (SELECT MAX(level) FROM q) - level AS level
    FROM q
)
INSERT INTO table1
SELECT
    (SELECT MAX(ID) FROM Table1) + level + 1 AS ID,
    CASE WHEN level = 0 THEN -1
         ELSE (SELECT MAX(ID) FROM Table1) + level
    END AS ParentGroupID,
    SomeValue + '-cloned'
FROM    q2

Result when run on your test data:

ID  ParentGroupID  SomeValue  
1   -1             a          
2   1              b          
3   2              c          
4   -1             a-cloned
5   4              b-cloned
6   5              c-cloned