= Solution
Projection onto the subring in part (b), along the two-sided <ideal>, is the <Brauer homomorphism>:
$$
\boxed{\operatorname{Br}_D:(kG)^D\to kC_G(D),\qquad
\sum_x a_xx\longmapsto\sum_{x\in C_G(D)}a_xx.}
$$
It is a unital <algebra> homomorphism because it is the quotient by that <ideal> followed by the displayed subring identification.
Put $N=N_G(D)$. The projection is $N$-equivariant, since $N$ preserves the <centralizer>. Its restricted kernel is
$$
\boxed{\ker(\operatorname{Br}_D|_{(kG)^N})
=(kG)^N\cap\sum_{E<D}\operatorname{Tr}_E^D((kG)^E).}
$$
Equivalently it is spanned by the sums of the $N$-conjugation orbits on $G$ disjoint from $C_G(D)$. Normality of $D$ in $N$ ensures that an $N$-orbit either lies wholly inside that <centralizer> or wholly outside it.
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