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I'll describe some general strategies for constructing (nonabelian) groups without referring to them as symmetries of something. Whether the examples arise "in nature" or certain actions are "natural" is sometimes debatable - this is an analogue ambiguity in the question that may be difficult to remove. I think for the purposes of your Picard this discussion, regular representations should not qualify as "natural" actions, even though they are quite natural.

Extensions of other groups: Given two groups $H$ and $K$, pick some group that fits into an exact sequence $1 \to H \to G \to K \to 1$ by specifying some datum, homological or otherwise. You might claim that $G$ acts on the total space of a certain $H$-torsor over $K$, or on some induced representation of $H$, but this seems to come close to regular representations. Anyway, examples include:

  • Central extensions of linear groups, suggested by coudy. The "natural" ones often have natural actions on infinite dimensional spaces (cf. Weil representation), so picking out an example that satisfies the conditions of the question could be thorny.
  • String groups (3-connected extensions of compact simple Lie groups by $K(\mathbb{Z},3)$), suggested by Allen. You could reasonably claim that these groups naturally act on categories appearing in the WZW model, such as module categories of certain vertex algebras, but it is not obvious if you only saw a purely topological construction.
  • Random finite $p$-groups, e.g., constructed by taking field-valued points on iterated extensions of finite length Witt vector groups. Most finite groups seem to have this form.
  • Finite perfect groups - not very well understood outside simple groups and their central extensions.

    • Quotients of large groups by normal subgroups:take Taking a quotient tends to destroy an action. Examples:

  • Take a free group and start tossing in relations. Yes, this is group of symmetries of a certain 2-complex, but that 2-complex is built out of the regular representation. I mentioned groups defined by presentations and automatic groups in a previous version of this answer, and they fit in here nicely.
  • Take a higher-categorical group and consider its $\pi_0$. The invertible objects in a monoidal (but not braided or symmetric) category . The $\mathcal{C}$ form a 2-group, and their isomorphism classes form a the Picard groupthat may not be abelian.

    Groups that arise Mariano mentioned in geometric the comments that this group theory acts naturally on the category in a weak sense, but the honest symmetries are sometimes given as a central extension of the Picard group by their presentations$B(\operatorname{Aut} 1)$

    • Intrinsic properties: I don't have a good example of this, or but in principle, there could be a group in nature that was uniquely defined by more abstract specifications like automatasome property, but didn't have a natural action arising from that property. You One could argue that they naturally act on their Cayley graphssome of the finite simple groups constructed in the classification program were "found" by searching through possible centralizers of involutions and deducing consequent properties, but you could say in the same for end, almost all of the class groupgroups were explicitly constructed by viewing them as symmetry groups of combinatorial or linear-algebraic objects. One could argue that some of the constructions given computationally by explicit generating matrices (e.g., some Janko groups) are unnatural, and I might agree.
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    Here's an analogue of your Picard example: take the invertible objects in a monoidal (but not braided or symmetric) category. The isomorphism classes form a group that may not be abelian.

    Groups that arise in geometric group theory are sometimes given by their presentations, or by more abstract specifications like automata. You could argue that they naturally act on their Cayley graphs, but you could say the same for the class group.