Random walks on a finite group and the Frobenius-Schur theorem

We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a substitution $g^2 \in G$ . Then the set of squares of elements of the group $G$ be a carrier of a probability $P(g)=\frac{r(g)}{|G|}\ (g \in G)$ , where $r(g)$ is a number of elements $h \in G$ such that $h^2 = g$ . Using well-known Frobenius-Schur theorem we find speed of convergence of $n$-fold convolution of $P$ to the uniform probability $U$ and conditions for the convergence.