= Solution
Partition the <conjugacy class> into its orbits under conjugation by $N$. Since none of its elements centralizes $N$, each orbit has size $[N:C_N(x)]>1$, a power of $p$. All elements of such an orbit have the same image in $G/N$, because conjugation by an element of $N$ does not alter an $N$-coset. Its orbit sum therefore has image
$$
[N:C_N(x)]\,\tau(x)=0
$$
in characteristic $p$. Summing over the orbits proves \b[$[\mathcal C]\in\ker\tau$]. This does not require $G=NC_G(N)$; that additional assumption is used in the block and Cartan conclusions proved in the root Solution.
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