Let’s define the set of *outer-commutator group words* $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence:

$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$

$$\forall u, v \in OC \text{ } [u, v] \in OC$$

Let’s call a group variety *outer commutator variety* iff it can be defined by a single outer commutator identity. Examples of outer commutator varieties include the varieties of

a) all $n$-step nilpotent groups (defined by $[…[[x_0, x_1], x_2]… x_n]$)

b) all $n$-Engel groups (defined by $[…[[x_0, x_1], x_1]… x_1]$)

c) all $n$-step soluble groups (defined by $[…[[x_0, x_1], [x_2, x_3]]…, ...[[x_{n-1}, x{n-2}],[x_{n-1}, x_n]]…]$

d) the trivial group (defined by $x_1$)

e) all groups (defined by $[x_1, x_1]$)

Also, the class of outer commutator varieties is closed under variety product.

There are three theorems proved by Eugeny Khukhro and Natalya Makarenko about outer commutator varieties:

**First Khukhro-Makarenko Theorem**

Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $H \in \mathfrak{U}$ and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $N \in \mathfrak{U}$.

**Second Khukhro-Makarenko Theorem**

Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $|V_\mathfrak{U}(H)| < \infty$ and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $|V_\mathfrak{U}(H)| < \infty$.

**Third Khukhro-Makarenko Theorem**

Suppose $\mathfrak{U}$ is an outer commutator variety, $G$ is a group and exists such subgroup $H \leq G$, that $V_\mathfrak{U}(H)$ is locally finite and $[G:H] < \infty$. Then exists a subgroup $N \leq G$, such that $N$ is characteristic $G$, $[G:N] < \infty$ and $V_\mathfrak{U}(H)$ is locally finite.

Here $V_\mathfrak{U}$ stands for the corresponding verbal subgroup.

For the First Khukhro-Makarenko theorem, there is an upper bound proved by Anton Klyachko and Yulia Melnikova:

If $\mathfrak{U}$ is defined by an outer commutator group word $w$, then

$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!)$$

where

$$f(x) = x(x+1)$$

$$f^{(n)}(x) = \begin{cases} x & \quad n = 0 \\ f(f^{(n-1)}(x)) & \quad n > 0 \end{cases}$$

$$d(x_i) = 1$$

$$d([v, u]) = d(v) + d(u)$$

For the conditions of the second Khukhro-Makarenko theorem a similar bound can be derived:

$$ln([G:N]) \leq f^{(d(w)+1)}([G:H]!) + ln(|V_\mathfrak{U}(H)|)$$

My question is:

Are there any similar upper bounds for $[G: N]$ under conditions of the Third Khukhro-Makarenko theorem?

Personally, I failed to find anything for that third case.

Yesterday, I asked this question on MSE and was advised to re-ask it here.