Let $F_n$ be a free group of finite rank $n$ and let $V$ be a verbal subgroup of $F_n.$ In a "nice" situation one could expect that

$(*):$ the group $\mathrm{Aut}_V(F_n) \le \mathrm{Aut}(F_n)$ whose elements fix each element of $F_n$ mod $V$ acts transitively on the set of all primitive elements from the coset $xV$ of a primitive element $x$ of $F_n.$

I have two questions related to the situation above.

1) Is the property $(*)$ true for the verbal subgroups $V=V([x_1,\ldots,x_m])=\gamma_m(F_n),$ i.e. for terms of the lower cenral series of $F_n?$ (Asked this question on math.stackexchange, got no answers).

Most likely, the answer to 1) is affirmative, and might be obtained from the classic description of the generators of the group $\mathrm{IA}(F_n)$ of IA-automorphisms of $F_n$ due to Magnus. I just don't see how, and being not versed in German I am not able to read the original Magnus' proof. Any comments on the plan etc. of Magmus' proof are welcome.

2) Are there examples of verbal subgroups $V$ for which the property $(*)$ is not true?