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 h is 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 [Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], Gould, Iskra, and Tsinakis write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined". Here, [16] is Tamura and Shafer's [Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on this paper of Tamura and Shafer, but to no avail. So, I don't know exactly what they prove inside it. But it seems from McAlister's zbMath review of the same article 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?