In [Ar68], M. Armstrong proved a beautiful result describing fundamental groups of quotient spaces. In this paper we prove an analogue of Armstrong's theorem in the setting of $F$-divided [dS07] and essentially finite [No76] fundamental group schemes.
"On the fundamental group schemes of certain quotient varieties." Tohoku Math. J. (2) 73 (4) 565 - 595, 2021. https://doi.org/10.2748/tmj.20200727