Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth commutative group scheme over $k$. Suppose now that $B_k$ is an abelian variety over $k$ that is a quotient of $A_k$.

Does there exist an abelian scheme $B$ over $R$ and a morphism $A\rightarrow B$ of smooth commutative $R$-groups whose base change to $k$ is the quotient mapping $A_k \rightarrow B_k$?

The answer is NO, I'm fairly sure, as the generic fiber of $A$ could be simple with $A_k$ an extension of an abelian variety by a torus. Let us therefore assume that $B_k$ can be lifted up to isogeny, i.e. that there exists $C$ an abelian scheme over $R$ such that 1) $C_k$ is isogenous to $B_k$ and 2) There exists a map of smooth groups $A\rightarrow C$ over $R$ whose base change to $k$ is the quotient map $A_k\rightarrow B_k$ followed by the isogeny $B_k\rightarrow C_k$. With this added assumption, is the answer to the question above still NO?

I'm inclined to think that this is the case, but can't immediately convince myself of this.