SE-SKOLORNAS 2020 Problem 5

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$ + ...+ ${p-1 \choose p-3} a^{p-5} .$