Pulling data from different sheets into a summary sheet

Yes there are ways but more details might be helpful or a sample sheet without any personal or private information.  In concept:

a) I would say and recommend that having a 'master sheet' with all your data input and then pulling subsets (pivot tables or filtered outputs) would be an easier and more efficient approach.  

b) you can create a combined output using VSTACK or something similar.  For example:

=VSTACK( EXPAND([names-in-A], , 2, "A"), EXPAND([names-in-B], , 2, "B"), EXPAND([names-in-C], , 2, "C"), EXPAND([names-in-D], , 2, "D"), EXPAND([names-in-E], , 2, "E") )

The EXPAND will create a second column with the corresponding group letter while VSTACK will stack all those ranges together.

Alternative, more automated options that could cycle across sheets could be done (e.g. you could use REDUCE to cycle over a range of sheet names and use INDIRECT to create each reference), but if you just have these 5 sheets this is probably easier and more efficient.  I would still recommend considering re-doing the structure to have all the data in a master table and then use other sheets to show filtered lists or "reports' accordingly.