What can we learn about a group by studying its monoid of subsets?

I believe it is always true that $M(G)\cong M(H)$ implies $G\cong H$. Because let $\phi:\ M(G)\rightarrow M(H)$ be a monoid isomorphism. Note that the only elements $x\in M(G)$ such that there exists $y\in M(G)$ with $xy=1$ are the one-element subsets of $G$. That is, the invertible elements in $M(G)$ are the one-element subsets of $G$; and the monoid product in $M(G)$, restricted to these one-element subsets of $G$, is just the group product of $G$. Thus $\phi$ restricted to these invertible elements induces an isomorphism between $G$ and $H$.

It turns out that there has been a large amount of research on the semigroups of all (finite/non-empty) subsets of a semigroup. For some reason the authors seem to prefer to consider only the non-empty subsets and I will stick to this convention in this post.

Since the question has been favorited by five people who aren't myself, I understand that the results obtained may be of interests to the community. I will post some things that I have found, especially terminology and references so anyone can do more searching by themself.

The semigroup of all non-empty subsets of a semigroup $S$ is called the power semigroup of $S,$ or the global of $S,$ and denoted by $\mathcal{P}(S).$ Semigroups $S_1,S_2$ are said to be globally isomorphic iff $\mathcal P(S_1)\cong \mathcal P (S_2).$ A class of semigroups $\mathscr C$ is said to be globally determined iff for any semigroups $S_1,S_2$ from $\mathscr C,$ if $S_1,S_2$ are globally isomorphic, then they are isomorphic.

The fact that the class of groups is globally determined was first noted by J. Shafer in /5/ and the reasoning was of course like Steve D's. The question whether the class of all semigroups is globally determined was posed in the 1960s and answered in the negative by E.M. Mogiljanskaja in /4/. The question has a positive answer for the class of full transformation semigroups (even more, any semigroup globally isomorphic to a full transformation semigroup must be isomorphic to it). This was proved by Vazenin in /6/. M. Gould and J. A. Iskra proved in /1/ that the class of finite simple semigroups is globally determined. It was later extended by Tamura to the classes of completely simple and completely 0-simple semigroups (please see the last link to find the definitions of these terms).

Gould, Iskra and C. Tsinakis later proved in /2/ another result, which needs a definition.

Defintion. An element $x$ of a semigroup $S$ is called irreducible iff for any $y,z\in S$ such that $x=yz,$ we have $x\in\{y,z\}.$

Note that $x$ is irreducible iff $S\setminus\{x\}$ is a subsemigroup of $S.$ Tamura and Shafer earlier termed such elements "prime". The theorem says what follows.

Theorem. The class of all completely regular periodic monoids in which the identity element is irreducible is globally determined.

Finally, Y. Kobayashi proved in /3/ that the class of all semilattices (treated as algebraic structures) is globally determined.

References.

/1/ M. Gould, J.A. Iskra, Globally determined classes of semigroups, Semigroup Forum Vol. 28 (1984), 1-11.

/2/ M. Gould, J.A. Iskra, C. Tsinakis, Globals of completely regular periodic groups, Semigroup Forum Vol 29 (1984), 365-374.

/3/ Y. Kobayashi, Semilattices are globally determined, Semigroup Forum Vol. 29 (1984), 217-222.

/4/ E. M. Mogilianskaja, The solution to a problem of Tamura, Sbornik Naučnyh Trudov Leningrad. Gos. Ped. Inst.,"Modern Analysis and Geometry", (1972), 148-151.

/5/ J. Shafer, Note on Power Semigroups, Math. Japan. 12 (1967), 32.

/6/ Ju. M. Vazenin, On the global oversemigroup of a symmetric semigroup, Mat. Zap. Ural. Gos. Univ 9 (1874), 3-10.

A lot is known on the monoid $\mathcal{P}(G)$ of subsets of a finite group $G$ (also called a power group). For instance, two subgroups of $G$ are conjugate if and only if they are $\mathcal{J}$-equivalent in $\mathcal{P}(G)$. For more details, see

J.-É. Pin, PG = BG, a success story, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (éd.), 33-47, Kluwer academic publishers, (1995). http://www.irif.fr/~jep/PDF/BGPG.pdf