SQL Server 2014

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create table periods (
   PeriodId INT,
   PeriodStart DATE,
   PeriodEnd DATE
);

INSERT INTO Periods (PeriodId, PeriodStart, PeriodEnd)
VALUES ('202005', '2020-05-01', '2020-05-31')
      ,('202006', '2020-06-01', '2020-06-30');
2 rows affected

create table Activities (
   ActivityId INT,
   UserId VARCHAR(255),
   ActivityStart DATE,
   ActivityEnd DATE
);

INSERT INTO Activities (ActivityId, UserId, ActivityStart, ActivityEnd)
VALUES (1, 'A', '2020-05-20', '2020-06-05')
      ,(2, 'A', '2020-06-15', '2020-06-18')
      ,(3, 'B', '2020-06-10', '2020-06-25');
3 rows affected

DECLARE @PeriodId int = 202006;

DECLARE @ps date, @pe date;
SELECT @ps = PeriodStart, @pe = PeriodEnd FROM dbo.Periods
   WHERE PeriodId = @PeriodId;
   
;WITH dates(rn,dt) AS 
(
    SELECT 1, @ps UNION ALL SELECT rn + 1, DATEADD(DAY, rn, @ps) 
    FROM dates WHERE dt < @pe
),
groups(UserId, dt, ActivityId, grp) AS
(
  SELECT u.UserId, d.dt, r.ActivityId, 
    d.rn - DENSE_RANK() OVER (PARTITION BY u.UserId, r.ActivityStart ORDER BY d.dt)
  FROM dates AS d CROSS JOIN (SELECT DISTINCT UserId FROM dbo.Activities 
    WHERE @pe >= ActivityStart AND @ps <= ActivityEnd) AS u
  LEFT OUTER JOIN dbo.Activities AS r
  ON u.UserId = r.UserId
  AND d.dt >= r.ActivityStart AND d.dt <= r.ActivityEnd
)
SELECT PeriodId = @PeriodId, UserId, ActivityId,
  NewActivityStart = MIN(dt),
  NewActivityEnd   = MAX(dt)
FROM groups 
GROUP BY UserId, ActivityId, grp
ORDER BY UserId, NewActivityStart;


PeriodId
UserId
ActivityId
NewActivityStart
NewActivityEnd


202006
A
1
2020-06-01
2020-06-05

202006
A
null
2020-06-06
2020-06-14

202006
A
2
2020-06-15
2020-06-18

202006
A
null
2020-06-19
2020-06-30

202006
B
null
2020-06-01
2020-06-09

202006
B
3
2020-06-10
2020-06-25

202006
B
null
2020-06-26
2020-06-30

PeriodId	UserId	ActivityId	NewActivityStart	NewActivityEnd
202006	A	1	2020-06-01	2020-06-05
202006	A	null	2020-06-06	2020-06-14
202006	A	2	2020-06-15	2020-06-18
202006	A	null	2020-06-19	2020-06-30
202006	B	null	2020-06-01	2020-06-09
202006	B	3	2020-06-10	2020-06-25
202006	B	null	2020-06-26	2020-06-30