This is Exercise 3A.9 - (b) from M. Isaacs' "Finite Group Theory". It goes as follows:
Let $W = H \wr G$ be the regular wreath product and let $C \leq G$ be an arbitrary subgroup. Show that there exists some $b \in B$, where $B = \{f:G \to H\}$ is the base group of $W$, such that $C = C_G(b)$
As far as I understand, an element of $G$ centralizes an arbitrary function $b \in B$ if and only if $b(g^{-1}k) = b(k), \forall k \in G$, since conjugation by an element of $G$ amounts to applying the action of that element; i.e: $$g^{-1}bg = g \cdot b$$
I tried to get some intuition, so I assumed first that $H = 1$. In particular, there is only one function from $G$ to $H$: everyone gets mapped to $1$. But this means every element of $G$ centralizes every element of $B$ (because there is only one), so it seems the result cannot possibly be true... What did I get wrong?
Nowhere were the actions required to be faithful, so I don't see what's wrong with picking a trivial example such as this one...
Thanks in advance!