So, are you looking for something like this?

IF OBJECT_ID('dbo.GroupMembers', 'U') IS NOT NULL DROP TABLE dbo.GroupMembers;

CREATE TABLE dbo.GroupMembers
(
    GroupCode VARCHAR(1) NOT NULL,
    GroupMember INT NOT NULL,
    StartDate DATE NOT NULL,
    EndDate DATE NULL
);

--Some of my groups have proper ranges for start and end dates
INSERT INTO dbo.GroupMembers VALUES ('A', 1, '20000101', '20170101');

--Some of my groups have were populated with a snapshot for every day
INSERT INTO dbo.GroupMembers VALUES ('B', 2, '20000101', '20000101');
INSERT INTO dbo.GroupMembers VALUES ('B', 2, '20000102', '20000102');
INSERT INTO dbo.GroupMembers VALUES ('B', 2, '20000103', '20000103');
INSERT INTO dbo.GroupMembers VALUES ('B', 2, '20000104', '20000104');

--Other groups are open ended and that is fine
INSERT INTO dbo.GroupMembers VALUES ('C', 3, '20000104', NULL);

--Some members may be removed and then re-added
INSERT INTO dbo.GroupMembers VALUES ('D', 4, '20170101', '20170101');
INSERT INTO dbo.GroupMembers VALUES ('D', 4, '20170102', '20170102');
INSERT INTO dbo.GroupMembers VALUES ('D', 4, '20170104', '20170104');
INSERT INTO dbo.GroupMembers VALUES ('D', 4, '20170105', '20170105');

SELECT * FROM dbo.GroupMembers;

WITH base AS (
SELECT
  [GroupCode]
  , [GroupMember]
  , [StartDate]
  , [EndDate]
  , rn = ROW_NUMBER() OVER (PARTITION BY [GroupCode], [GroupMember] ORDER BY [StartDate])
FROM
  dbo.GroupMembers
), Interim AS (
SELECT
  .[GroupCode]
  , .[GroupMember]
  , .[StartDate]
  , .[EndDate]
  , .[rn]
  , [GroupDate] = DATEADD(DAY,-.[rn],.[StartDate])
FROM
  base b
)
SELECT
  .[GroupCode]
  , .[GroupMember]
  , [StartDate] = MIN(.[StartDate])
  , [EndDate] = MAX(.[EndDate])
FROM
  [Interim] i
GROUP BY
  .[GroupCode]
  , .[GroupMember]
  , .[GroupDate];