Outer automorphism groups of nilpotent groups

Question

I am interested in the class of finitely generated nilpotent groups. It seems that many things are known about the automorphism groups of such groups. However, I could not find much information on their outer automorphism groups. The state of the art, according to a basic search, appears to be a paper of T. Fournelle from the '80s, from which the nontriviality of $\operatorname{Out}(G)$ follows for infinite finitely generated nilpotent $G$.

A classical result I am particularly interested in is that $\operatorname{Aut}{G}$ is linear (in fact, even the holomorph $G \rtimes \operatorname{Aut}(G)$ is linear, and more generally for $G$ virtually polycyclic, though this is a bit harder than the nilpotent case). An account may be found in Chapter 5 of Segal's book Polycyclic groups. My question is thus as follows

Let $G$ be a finitely generated nilpotent group. Is $\operatorname{Out}(G)$ linear?

In particular, are there known examples for which the answer to the above is negative? And if not, are there reasons why one should expect the result to go one way or the other? There is also perhaps the related:

Let $G$ be a linear group, $N$ a normal finitely generated nilpotent subgroup. Is the quotient $G/N$ linear? What if $N$ is abelian?

This is certainly a much stronger statement, and an affirmative answer would answer the first question. I would be somewhat surprised if it were true in general, at least in the nilpotent case; counterexamples would be appreciated, should they exist.

References:

Fournelle, Thomas A., ZBL0424.20033.

Answer 1

Following the useful reference provided by YCor in the comments above, I found a paper of Segal from 1989 that contains the original answer to the first question: $\operatorname{Out}(G)$ is in fact $\mathbb{Z}$-linear. This is generalised to virtually polycyclic groups in a paper of Wehrfritz, appearing shortly after. The work of Baues and Grunewald asserts the (much more powerful) property that $\operatorname{Out}(G)$ is arithmetic for $G$ virtually polycyclic.

Remarkably, this paper also gives an answer to the latter question if linear is replaced with $\mathbb{Z}$-linear. He proves:

Let $N$ be a normal solvable-by-finite subgroup of $\mathbb{Z}$-linear group $G$. Then $G/N$ is also $\mathbb{Z}$-linear.

Combined with the other comment indicating that this fails even when $\mathbb{Z}$ is replaced with $\mathbb{Z}[1/p]$, this gives a fairly complete answer to the second question as well. I guess this is a good lesson that one should always try to search for results more general than your current setting (especially when they admit such an obvious generalisation!).

References:

Segal, Dan, On the outer automorphism group of a polycyclic group, Group theory, Proc. 2nd Int. Conf., Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 265-278 (1990). ZBL0703.20034.

Wehrfritz, B. A. F., ZBL0819.20036.