T7rkelli's modification for $(T\normal G)$-towers

Let $G\subset S_n$ be transitive, let $T\normal G$, let $(L/K,\iota_B)$ be a $B$-extension, and let $\pi:G_K\rightarrow G$ be a homomorphism with $\pi\equiv \iota_B\mod T$. Write $N(L/K,T\normal G;X)$ for the counting function and $c'(K,T(\pi))$ for the associated constant. T7rkelli's modification for $(T\normal G)$-towers. Then

$$N(L/K,T\normal G;X) \sim c'(K,T(\pi)) X^{1/a(T)} (\log X)^{B(K,T(\pi))-1}.$$

This is the proposed generalized asymptotic form of T7rkelli's modification for towers with normal subgroup $T\normal G$, refining the predicted logarithmic exponent through the modified invariant $B(K,T(\pi))$. The parser supplies no evidence resolving the statement.