Are abelian varieties unobstructed? That is, given an abelian scheme $X \to \mathrm{Spec}R $ for $R$ a local artinian ring and $\mathrm{Spec} R \to \mathrm{Spec} S $ a nilpotent thickening, can we extend $X$ to an abelian scheme over $\mathrm {Spec }S $?

This assertion was made somewhat quickly in an article of Katz I was trying to read, at the bottom of p. 144. Mumford shows in his GIT book that it's enough to deform $X$ together with the identity section $\mathrm{Spec} R \to \mathrm{Spec } X$, and the abelian structure follows. I don't see how to get that, though.