I have a table like this:
Name   activity  time
user1  A1        12:00
user1  E3        12:01
user1  A2        12:02
user2  A1        10:05
user2  A2                
        Test setup:
CREATE TEMP TABLE t (name text, activity text, time time);
INSERT INTO t values
 ('user1', 'A1', '12:00')
,('user1', 'E3', '12:01')
,('user1', 'A2', '12:02')
,('user2', 'A1', '10:05')
,('user2', 'A2', '10:06')
,('user2', 'A3', '10:07')
,('user2', 'M6', '10:07')
,('user2', 'B1', '10:08')
,('user3', 'A1', '14:15')
,('user3', 'B2', '14:20')
,('user3', 'D1', '14:25')
,('user3', 'D2', '14:30');
Your definition:
activity from group B always takes place after activity from group A.
.. logically implies that there is, per user, 0 or 1 B activity after 1 or more A activities. Never more than 1 B activities in sequence.
You can make it work with a single window function, DISTINCT ON and CASE, which should be the fastest way for few rows per user (also see below):
SELECT name
     , CASE WHEN a2 LIKE 'B%' THEN a1 ELSE a2 END AS activity
     , CASE WHEN a2 LIKE 'B%' THEN a2 END AS next_activity
FROM  (
   SELECT DISTINCT ON (name)
          name
        , lead(activity) OVER (PARTITION BY name ORDER BY time DESC) AS a1
        , activity AS a2
   FROM   t
   WHERE (activity LIKE 'A%' OR activity LIKE 'B%')
   ORDER  BY name, time DESC
   ) sub;
An SQL CASE expression defaults to NULL if no ELSE branch is added, so I kept that short.
Also assuming time is defined NOT NULL. Else, you might want to add NULLS LAST. Why?
(activity LIKE 'A%' OR activity LIKE 'B%') is more verbose than activity ~ '^[AB]', but typically faster in older versions of Postgres. About pattern matching:
That's actually possible. You can combine the aggregate FILTER clause with the OVER clause of window functions. However:
The FILTER clause itself can only work with values from the current row.
More importantly, FILTER is not implemented for pure window functions like lead() or lag() in Postgres 9.6 (yet) - only for aggregate functions.
If you try:
lead(activity) FILTER (WHERE activity LIKE 'A%') OVER () AS activity
Postgres will tell you:
FILTER is not implemented for non-aggregate window functions
About FILTER:
(For few users with few rows per user, pretty much any query is fast, even without index.)
For many users and few rows per user, the first query above should be fastest. See the linked answer above about index and performance.
For many rows per user, there are (potentially much) faster techniques, depending on other details of your setup:
select      distinct on(name) name,activity,next_activity
from       (select name,activity,time
                  ,lead(activity) over (partition by name order by time) as next_activity
            from   t
            where  left(activity,1) in ('A','B')
            ) t
where       left(activity,1) = 'A'
order by    name,time desc