SQL Server 2019

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create table t (name varchar(10), date_from date, date_to date);
insert into t (name, date_from, date_to) values
  ('A', '2022-01-03', '2022-01-03'),
  ('A', '2021-12-29', '2021-12-31'),
  ('A', '2021-12-28', '2021-12-28'),
  ('A', '2021-12-27', '2021-12-27'),
  ('A', '2021-12-23', '2021-12-24'),
  ('A', '2021-11-08', '2021-11-09');
6 rows affected

select name, min(date_from) as date_from, max(date_to) as date_to
from (
  select *, sum(inc) over(partition by name order by date_to) as grp
  from (
    select *,
      case when lag(ext_to) over(partition by name order by date_to) = date_from
           then 0 else 1 end as inc
    from (
      select *,
        case when (datepart(weekday, date_to) = 6) 
             then dateadd(day, 3, date_to)
             else dateadd(day, 1, date_to) end as ext_to
      from t
    ) x
  ) y
) z
group by name, grp
order by name, grp


name
date_from
date_to


A
2021-11-08
2021-11-09

A
2021-12-23
2022-01-03