I am interested in the solutions of the equation $2^{q-1} \equiv q \pmod {p} $ where $p=4q^2+1$ for an odd prime $q$.

So far the only solution I found by trial and error is $q=193$ but I don't know how to realize it as a solution. I'd like to know if there are other solutions.

By relaxing the condition on $q$ one gets two more solutions: $q=1,2$. It would also be interesting to see if there are other positive integer solutions.