It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian variety over $W$, the ring of Witt vectors has a group structure. This prompts the following question: Does this group structure extend to the formal moduli of nonordinary abelian varieties? and if not, what is the biggest class of abelian varieties, for which, a group structure on the formal moduli exists?

I cannot comment yet, so I will add the following remark as an answer. I do not think that this answers your question, but at least it is a case where a group structure does not exist, yet some generalization does: In SerreTate theory for moduli spaces of PELtype, Ann. scient. de l'Ec. Norm. Sup. 37 (2004), 223269, (arXiv:math/0203288v2), Ben Moonen looks at this question for abelian varieties (equivalently: $p$divisible groups) with additional structure in the $\mu$ordinary case. 

