Number of $2\times 2$ matrices $X$ such that $X^2 \equiv I \mod p$

Question

From what I was able to find. The number of $2\times 2$ invertible matrices over $p$ is $(p^2-1)(p^2-p)$. Since I only need squares to be equivalent to $I$, the number of matrices reduces to $(p^2-1)(p-1)$, but I cannot understand what to do with square. I guess it should be such a matrix that should be equivalent to $I$ too. Does this mean that the amount of $X^2$ is equal to the amount of $X$?