SQL Server 2017

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select @@version;


(No column name)


Microsoft SQL Server 2017 (RTM-CU21) (KB4557397) - 14.0.3335.7 (X64) 
 Jun 12 2020 20:39:00 
 Copyright (C) 2017 Microsoft Corporation
 Express Edition (64-bit) on Windows Server 2019 Standard 10.0 <X64> (Build 17763: ) (Hypervisor)



create table t(timestamp1 datetime, category varchar(10));
insert into t values('2018-10-01 04:06:12','Cat1')
insert into t values('2018-10-01 05:07:18','Cat1')
insert into t values('2018-10-01 05:07:19','Cat1')
insert into t values('2018-10-01 05:07:20','Cat1')
insert into t values('2018-10-01 06:09:29','Cat1')
insert into t values('2018-10-01 07:24:12','Cat2')
insert into t values('2018-10-01 07:30:43','Cat2')
insert into t values('2018-10-01 07:59:13','Cat2')
insert into t values('2018-10-01 08:02:15','Cat2')
insert into t values('2018-10-01 10:09:25','Cat2')
insert into t values('2018-10-01 11:13:42','Cat2')
11 rows affected

with raw_data
  as(select timestamp1
            ,category
            ,case when datediff(mi,lag(timestamp1) over(partition by category order by timestamp1),timestamp1) >5 
                    or lag(timestamp1) over(partition by category order by timestamp1) is null
                  then row_number() over(partition by category order by timestamp1)                  
              end as grps_of_5
       from t  
     )
   ,curated_data
      as (select max(grps_of_5) over(partition by category order by timestamp1) as max_val
                 ,x.*
            from raw_data x
           )
 select category,count(distinct max_val) as cnt
   from curated_data
group by category   


category
cnt


Cat1
3

Cat2
5


Warning: Null value is eliminated by an aggregate or other SET operation.