If I understood you correctly, your problem isn't really how to group by a date: GROUP BY [ASSIGNED] takes care of that. I'm sure you also have no problem creating a WHERE clause to select only rows with [ASSIGNED] dates in the specified range.

Your problem is you do not want to count quantities of assigned items where they are beyond the end date. For that you need a CASE expression that only returns the value of quantities for dates prior to [ENDS].

select Assigned, Item, SUM(case when [Assigned] <= [Ends] then Qty else 0 end) as QTY

from @t

join blah blah blah

where blah blah blah

group by Item, [assigned]

order by Item, [assigned]