  We make a connection between the subgroup membership and identity problems
for matrix groups and extended finite automata. We provide an alternative proof
for the decidability of the subgroup membership problem for $ 2 \times 2 $
integer matrices. We show that the emptiness problem for extended finite
automata over $ 4 \times 4 $ integer matrix semigroups is undecidable. We prove
that the decidability of the universe problem for extended finite automata is a
sufficient condition for the decidability of the subgroup membership and
identity problems.
