The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.

Optimal indexes for this query:

ALTER TABLE `Activities` ADD INDEX `activities_idx_isactive` (`IsActive`);
ALTER TABLE `ActivityDetails` ADD INDEX `activitydetails_idx_isactive_activityid` (`IsActive`,`ActivityId`);

The optimized query:

SELECT
        Activities.BranchId,
        Activities.Type,
        ActivityDetails.ItemId,
        sum(ActivityDetails.Qty) 
    FROM
        Activities 
    JOIN
        ActivityDetails 
            ON Activities.Id = ActivityDetails.ActivityId 
            AND ActivityDetails.IsActive = 1 
    WHERE
        Activities.IsActive = 1 
    GROUP BY
        Activities.BranchId,
        Activities.Type,
        ActivityDetails.ItemId 
    ORDER BY
        NULL