Examples of "natural" finitely generated groups with an undecidable conjugacy problem I am looking for natural groups with undecidable conjugacy problem. By natural, I mean that the word problem should be decidable, and the group should be given by some natural action. I know that $\mathbb{Z}^d \rtimes F_m$ (with a suitable action of $F_m$) has undecidable conjugacy problem. That's very nice, but I'd like to know other examples. I do not care about finite presentation, and I'm also fine with the group being a f.g. subgroup of something natural and geometric, which maybe simplifies things. A concrete case I was not able to resolve is whether all f.g. subgroups of right-angled Artin groups have decidable conjugacy problem.
Šunić, Zoran; Ventura, Enric, The conjugacy problem in automaton groups is not solvable., J. Algebra 364, 148-154 (2012). ZBL1261.20034.
 A: Chuck Miller in [Miller, Charles F., III On group-theoretic decision problems and their classification. Annals of Mathematics Studies, No. 68. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971] proves the following two rather nice and natural examples.
Theorem III.10. The free product of two free groups with finitely generated amalgamation can have unsolvable conjugacy problem. Further, the finitely presented HNN extension of a free group can have unsolvable conjugacy problem.
(Note that Miller calls HNN extensions 'Strong Britton extensions').
Now by Bass-Serre theory, there is a natural action of an amalgamated free product/HNN on the associated Bass-Serre tree, which should satisfy your "natural action" criterion.

Edit: The result mentioned by YCor can also be found in Miller's book.
Theorem III.23 The group $F_2 \times F_2$ has a finitely generated subgroup with undecidable conjugacy problem.
An important side remark, however, is that $F_2 \times F_2$ itself has decidable conjugacy problem, as do all RAAGs, in linear time. See [Crisp, John; Godelle, Eddy; Wiest, Bert; The conjugacy problem in subgroups of right-angled Artin groups. J. Topol. 2 (2009), no. 3, 442–460.].
