First law of cubology for Rubik groups of oriented maps

Let $M$ be an oriented map with vertex set $V(M)$ and edge set $E(M)$, and let $H_1(M)$, $H_2(M)$, and $H_3(M)$ be the kernels of the successive homomorphisms in the decomposition of $Rubik(M)$:

$$H_1(M)=\operatorname{Ker}(Rubik(M)_{corner, side\,\,edge}\to Rubik(M)_{corner, edge}),$$ $$H_2(M)=\operatorname{Ker}(Rubik(M)_{corner, edge}\to Rubik(M)_{corner}),$$ $$H_3(M)=\operatorname{Ker}(Rubik(M)_{corner}\to Rubik(M)_{vertex}).$$

Here $M$ has $v$ vertices and $e$ edges. First law of cubology. The following is conjectured:

$$H_1(M)\cong {\mathbb{Z}}_2^{|E(M)|-1},\qquad H_2(M)\cong A_{|E(M)|},$$ $$H_3(M)\cong {\mathbb{Z}}_3^{|V(M)|-1},$$

and

$$Rubik(M)_{vertex}\cong A_{|V(M)|}$$

if all faces of $M$ have odd size, while

$$Rubik(M)_{vertex}\cong S_{|V(M)|}$$

otherwise. This conjectural description aims to determine the successive kernel factors and the induced permutation group on vertices of the Rubik group of an oriented map. The supplied context proves only subgroup inclusions, not the asserted isomorphisms, and gives no resolution of the conjecture.