TITLE: What are the auto-equivalences of the category of groups?
QUESTION [17 upvotes]: My question is motivated by Are the inner automorphisms the only ones that extend to every overgroup?
What are the auto-equivalences of the category of groups? What kind of structure do they form?
There are certainly some not-so-trivial examples (which are still trivial if one just focus on a single group), e.g. the functor send an abstract group G to the group G^op with the same underlying set with multiplication g,h ---> hg, where hg is the product of h and g in G. [The notation G^op will make sense if you think G as the category BG]

REPLY [13 votes]: Just to make this a little more visible: for a proof that every autoequivalence of the category of groups is naturally isomorphic to the identity, see page 31 of Peter Freyd's book "Abelian Categories".