I am sorry to bother you again, but there are some things I still do not get: if X is an abelian variety and Y its (singular) kummer variety (char$k\neq 2$) then the action is not free and the quotient map is not flat. If I have a G−sheaf on $X$ then I canconsider the sheaf $\pi_*F^G$ If I take the fiber at a a singular point $\overline{x}$ what do I obtain? I believed to get $\pi_*(F(x))^G$ but I do not think this is the same as $(\pi_*(F)(\overline{x}))^G$ (that is quoting t3suji the 'invariants in the fiber of the direct image'). What I am doing wrong? Thanks again and again sorry!

Wow! Thank you a lot for your answers/comments! That was really helpful. Now I will try to do my homework and proving the case in which the action is not free.. thank you again best wishes Stgermain

Yes, you are right! A friend of mine told me the same thing, yesterday, thus I removed the tag! Itagget it a s a soft question because after t3suji answered me I understood it was really easy (and I should have known it form the beginning :'(). Thank you for your comment!

Thank you very much for your comment/answer. I really had not understood that the canonical bilinear pairng $e_2:X\times\widehat X$ was the same of what Mumford dentoes as $e_*^P(\cdot,\cdot)$ with $P$ the Poincarè bundle. Thank you again for the time you dedicated to me Best Regards,