SQL Server 2019

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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

with dates as (
      select periodid, periodstart as dte, periodend
      from periods
      union all
      select periodid, dateadd(day, 1, dte), periodend
      from dates
      where dte < periodend
     )
select userid, activityid, min(dte), max(dte)
from (select d.dte, d.periodid, u.userid, a.activityid,
             row_number() over (partition by u.userid, a.activityid order by d.dte) as seqnum
      from dates d cross join
           (select distinct userid from activities) u left join
           activities a
           on a.userid = u.userid and
              a.activitystart <= d.dte and a.activityend >= d.dte
     ) da
group by userid, activityid, periodid, dateadd(day, -seqnum, dte)
order by userid, min(dte)


userid
activityid
(No column name)
(No column name)


A
null
2020-05-01
2020-05-19

A
1
2020-05-20
2020-05-31

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

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

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

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

B
null
2020-05-01
2020-05-31

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

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

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

userid	activityid	(No column name)	(No column name)
A	null	2020-05-01	2020-05-19
A	1	2020-05-20	2020-05-31
A	1	2020-06-01	2020-06-05
A	null	2020-06-06	2020-06-14
A	2	2020-06-15	2020-06-18
A	null	2020-06-19	2020-06-30
B	null	2020-05-01	2020-05-31
B	null	2020-06-01	2020-06-09
B	3	2020-06-10	2020-06-25
B	null	2020-06-26	2020-06-30