Orders and Primitive Roots · Superchallenge problem

Modulo the prime $61$, call $a$ (with $1 \le a \le 60$) a $k$-th power if $x^k \equiv a \pmod{61}$ has a solution. For how many $a$ with $1 \le a \le 60$ is $a$ a $k$-th power for exactly two of the four values $k = 2, 3, 4, 5$?