You can count the distinct ProjIds, and then group by GroupId. In other words this query only counts each GroupId+ProjId combination once, no matter how many times the combination appears.

SELECT
        GroupID, 
        COUNT(DISTINCT ProjId)
    FROM
        Immerse
    GROUP BY GroupID

When asking a question it's really helpful to provide the DDL and DML for your demo data:

DECLARE @Immerse TABLE (ProjID INT, GroupID VARCHAR(3));
INSERT INTO @Immerse (ProjID, GroupId) VALUES
(1, 'AAA'), (2, 'BBB'), (3, 'AAA'), (4, 'CCC'), (4, 'CCC'), (4, 'CCC');

In this case it looks like a quick self join might be the way to go:

SELECT i1.GroupID, COUNT(i2.GroupID) AS NumberOfProjects, STRING_AGG(i2.ProjID,', ') AS OtherProjects
  FROM @Immerse i1
    LEFT OUTER JOIN @Immerse i2
      ON i1.GroupID = i2.GroupID
      AND i1.ProjID <> i2.ProjID
GROUP BY i1.GroupID;