SQL Server 2019

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CREATE TABLE observations
    ([Group] varchar(1), [Start] varchar(10), [End] varchar(10))
;
    
INSERT INTO observations
    ([Group], [Start], [End])
VALUES
    ('A', '2021-01-01', '2021-04-05'),
    ('A', '2021-01-01', '2021-06-05'),
    ('A', '2021-03-01', '2021-06-05'),
    ('B', '2021-06-13', '2021-08-05'),
    ('B', '2021-06-13', '2021-09-05'),
    ('B', '2021-07-01', '2021-09-05'),
    ('C', '2021-10-07', '2021-10-17'),
    ('C', '2021-10-07', '2021-11-15'),
    ('C', '2021-11-12', '2021-11-15')
;
9 rows affected

DELETE
 o1
 FROM observations o1 
 INNER JOIN(SELECT MIN([Start]) minstart , [Group] FROM observations GROUP BY [Group] ) o2
 ON o1.[Group] = o2.[Group] AND o1.[Start] <> o2.minstart
3 rows affected

SELECT *
 FROM observations


Group
Start
End


A
2021-01-01
2021-04-05

A
2021-01-01
2021-06-05

B
2021-06-13
2021-08-05

B
2021-06-13
2021-09-05

C
2021-10-07
2021-10-17

C
2021-10-07
2021-11-15

Group	Start	End
A	2021-01-01	2021-04-05
A	2021-01-01	2021-06-05
B	2021-06-13	2021-08-05
B	2021-06-13	2021-09-05
C	2021-10-07	2021-10-17
C	2021-10-07	2021-11-15