Let $X$ be a finite CW-complex and $A$ an abelian group, and consider the space $Maps(X,K(A,n))$ of continuous maps from $X$ to $K(A,n)$ endowed with the compact-open topology, so that it represents the functor $Y\mapsto C(X\times Y,K(A,n))$. Let $Maps_0(X,K(A,n))$ be the path-connected component of $Maps(X,K(A,n))$ corresponding to the zero in $H^n(X,A)$, and let $*$ any point in $Maps_0(X,K(A,n))$. I'm interested in the fundamental group $\pi_1( Maps_0(X,K(A,n)),*)$.

i) is it abelian?

ii) does it act trivially on $\pi_i( Maps_0(X,K(A,n)),*)$ for $i\geq 1$?

If the answer to these two were "yes", then one would have that for $X$ a closed compact connected smooth oriented manifold of dimension $k\lt n$, the Postnikov tower of $Maps(X,K(\mathbb{Z},n))$ begins with $K(\mathbb{Z},n-k)$, so that there would be a canonical $(n-k-2)$-gerbe on the (moduli) space of $(n-2)$-gerbes on $X$