Frenkel, Lepowsky, Meurman constructed the vertex operator algebra (VOA) $V^\natural$ as a chiral orbifold CFT whose target space is $\mathbb{R}^{24}/\Lambda/\mathbb{Z}_2$. (Here the last $\mathbb{Z}_2$ is a group of order two which acts by sending $\vec x\to -\vec x$, and not the ring of 2-adic integers. Sorry for using physics notation.)

This construction makes (almost) manifest that the said VOA has $2^{1+24}_{+}\cdot Co1$ as part of its symmetry group. In addition, there is an operation $\sigma$ which mixes the twisted sector and the untwisted sector. Then, adding $\sigma$ to $2^{1+24}_+\cdot Co1$ gives rise to the monster simple group $M$. Then $M$ has another involution $z$ (possibly conjugate to $\sigma$) such that $2^{1+24}_+\cdot Co1$ is the centralizer of $z$, $C_G(z)$. (Honestly I don't really understand the construction, but that's the story I've heard.)

Now, there are many other simple groups $G$ which has a similar structure, i.e. there is an involution $z\in G$ such that the centralizer $C_G(z)$ has the structure $2^{1+n}.H$.

Then my question is: is it always the case that there is a lattice $L$ of rank $n$ whose symmetry is $H$ (and $-1$) such that the VOA based on $\mathbb{R}^{n}/L/\mathbb{Z}_2$ has the symmetry $G$, given by adjoining an operation $\sigma$ mixing the twisted and the untwisted sector, to the part $2^{1+n}.H$ which exists almost by construction?

Update:

I realized now that Jeff Harvey already asked almost the same question a few months ago, see this MO question. I even made a comment in that thread; I completely forgot about that. You see, I'm only slowly digesting the interesting problem Jeff raised...

But of course the moderators can close my question as an exact duplicate. Sorry about that.