Fix a finite group $G$ and a connected manifold $M$. Let $Bun_G(M)$ denote the groupoid of principal $G$ bundles on $M$.
We know the classification theorem for principal G-bundles which states that $\pi_0(Bun_G(M)) = [M,BG]$ where $[M,BG]$ represents the homotopy classes of maps from $M$ to $BG$ and $BG$ is the classifying space of the group $G$. We can rewrite the above as: $$[M, BG] = \pi_0(\text{Maps}(M,BG))$$ where $Maps(M,BG)$ is the topological space of maps from $M$ to $BG$.
Given a principal $G$-bundle $P$, I am guessing that the automorphism group of $G$ bundles, $Aut(P)$ should be equal to $$\pi_1(\text{Maps}(M,BG),P)= \pi_0(\Omega_P \text{Maps}(M,BG))$$ where I am considering the classifying map of $P$ as a base point.
I think I should be able to simplify further and relate it to fundamental group of $M$. But I don't know how to proceed. Also, there might be mistakes in my arguments above. Please tell me if there are mistakes.