  We determine continuous bijections $f$, acting on a real interval into
itself, whose $k$-fold iterate is the quasi-arithmetic mean of all its
subsequent iterates from $f^0$ up to $f^n$ (where $0\le k\le n$). Namely, we
prove that if at most one of the numbers $k,n$ is odd, then such functions
consist of at most three affine pieces.
