AUSTRIAN-POLISH 1991 Problem 9

For a positive integer $n$ denote $A = \{1,2,..., n\}$ . Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$ . How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$ ?