CIIM 2017 Problem 3

Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$ ). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$