I'd like to search for flights departing either Fri PM or Sat AM, and returning either Sun PM or Mon AM.
In Google Flights, this requires 4 separate searches. Is there another site/tool that allows such a search?
On Matrix Airfare Search you have the option of searching for flexible dates. In my case I was able to search for flights from LHR to FRA leaving Friday and/or day before and coming back Monday on this day only - but you can change that to be the day before/after. You can also specify what time of day (early morning, midday, etc.)
https://travelgam.com/ is a site meant for exactly this purpose. Users can define start of weekend and end of weekend in terms of both day and time. Such as Friday late evening and Monday early morning respectively.
Outbound and Return times can be specified along with an option for "Direct flights only".