# Modular version of Mednykh's formula?

Let $$G$$ be a finite group and $$\Sigma_g$$ a closed Riemann surface of genus $$g$$. Then Mednykh's formula states

$$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma_g),G)\right|}{\left|G\right|} = \frac{1}{\left|G\right|^{\chi(\Sigma_g)}}\sum_{V}\left(\dim V\right)^{\chi(\Sigma_g)}$$

where the sum on the right is over irreducible complex representations of $$G$$.

Is there an analog in modular representation theory? For example we can fix the problem in the case $$g=0,1$$: the number of irreducible representations equals to number of $$p$$-regular conjugacy classes, the vector $$x=(\text{dim} V_i)^t$$ satisfies $$x^tCx=|G|$$ where $$C=D^tD$$ is the Cartan matrix.