Loosely speaking, a right group action of G on a set S is a map S × G → S mapping (s, g) → s·g, satisfying (s·g)·h = s·(g·h).
Suppose we define F(g)·s := s·g and we know that F(g·h) = F(h)·F(g) because F is a contravariant functor.
Now see that s·(g·h) = F(g·h)·s = (F(h)·F(g))·s = F(h)·(F(g)·s) = F(h)(s·g) = (s·g)·h and therefore, right actions are contravariant functors.
Suppose we define F(g)·s := s·g and we know that F(g·h) = F(h)·F(g) because F is a contravariant functor.
Now see that s·(g·h) = F(g·h)·s = (F(h)·F(g))·s = F(h)·(F(g)·s) = F(h)(s·g) = (s·g)·h and therefore, right actions are contravariant functors.
In this regard, I want to quote a line from Tom Leinster's Basic Category Theory: "That left actions are covariant functors and right actions are contravariant functors is a consequence of a basic notational choice: we write the value of a function f at an element x as f(x), not (x)f."