  Groups that can be approximated by finite groups have been the center of much
research. This has led to the investigations of the subgroups of metric
ultraproducts of finite groups. This paper attempts to study the dual problem:
what are the quotients of ultraproducts of finite groups?
  Since an ultraproduct is an abstract quotient of the direct product, this
also led to a more general question: what are the abstract quotients of
profinite groups? Certain cases were already well-studied. For example, if we
have a pro-solvable group, then it is well-known that any finite abstract
quotients are still solvable.
  This paper studies the case of profinite groups from a surjective inverse
system of non-solvable finite groups. We shall show that, for various classes
of groups $\mathcal{C}$, any finite abstract quotient of a pro-$\mathcal{C}$
group is still in $\mathcal{C}$. Here $\mathcal{C}$ can be the class of finite
semisimple groups, or the class of central products of quasisimple groups, or
the class of products of finite almost simple groups and finite solvable
groups, or the class of finite perfect groups with bounded commutator width.
  Furthermore, in the case of finite perfect groups with unbounded commutator
width, we show that this is NOT the case. In fact, any finite group could be a
quotient of an ultraproduct of finite perfect groups. We provide an explicit
construction on how to achieve this for each finite group.
