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We continue the study of the structure of general subgroups (in particularmaximal subgroups, also known as group $\mathcal{H}$-classes) of specialinverse monoids. Recent research of the authors has established that these canbe quite wild, but in this paper we show that if we restrict to special inversemonoids which are $E$-unitary (or have a weaker property we call$\mathcal{R}_1$-injectivity), the maximal subgroups are strongly governed bythe group of units. In particular, every maximal subgroup has a finite indexsubgroup which embeds in the group of units. We give a construction to showthat every finite group can arise as a maximal subgroup in an$\mathcal{R}_1$-injective special inverse monoid with trivial group of units.It remains open whether every combination of a group $G$ and finite indexsubgroup $H$ can arise as maximal subgroup and group of units.

Subgroups of $E$-unitary and $R_1$-injective special inverse monoids