Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved?