15.76 (2002)

If $\Theta$ is a variety of groups, then let $\Theta^0$ denote the category of all free groups of finite rank in $\Theta$. It is proved (G. Mashevitzky, B. Plotkin, E. Plotkin J. Algebra, 282 (2004), 490–512) that if $\Theta$ is the variety of all groups, then every automorphism of the category $\Theta^0$ is an inner one. The same is true if $\Theta$ is the variety of all abelian groups.
$\qquad$ a) Is this true for the variety of nilpotent groups of class 2?
$\qquad$ b) The same is true if $\Theta$ is the variety of all nilpotent groups of class $\leqslant d$, for $d \geqslant 2$ (A. Tsurkov, Int. J. Algebra Comput., 17 (2007), 1273–1281). Is this true for the varieties of solvable groups? of metabelian groups?

An automorphism $\varphi$ of a category is called inner if it is isomorphic to the identity automorphism. Let $s : 1 \to \varphi$ be a function defining this isomorphism. Then for every object $A$ we have an isomorphism $s_A : A \to \varphi(A)$ and for any morphism of objects $\mu : A \to B$ we have $\varphi(\mu) = s_B \mu s_A^{-1}$.