= Solution
Let $I$ be the <augmentation ideal> of $kN$, generated by the elements $n-1$. First prove its nilpotence. A nontrivial finite $p$-group has a central element $z$ of order $p$. Put $t=z-1$. In characteristic $p$, $t^p=z^p-1=0$, and
$$
kN/tkN\cong k(N/\langle z\rangle).
$$
Induction on $|N|$ says that the image of $I$ in this smaller <group algebra> is nilpotent. Thus $I^m\subseteq tkN$ for some $m$, and $I^{mp}\subseteq t^pkN=0$. The trivial <group> starts the induction. This proves <nilpotence of a p-group augmentation ideal>.
The quotient homomorphism kills exactly the differences of elements in each $N$-coset, so
$$
\ker\tau=I kG=kG I.
$$
Normality of $N$ makes $I$ stable under conjugation by $G$. Moving <group> elements past factors from $I$ therefore gives $(I kG)^r=I^r kG$. Consequently
$$
\boxed{\ker\tau\text{ is a nilpotent ideal of }kG.}
$$
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