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.

Reformulation

Consider the following theorem of Chevalley (see 9.2/1 of the book "Neron Models" by Bosch, Lutkebohmert and Raynaud):

Theorem: Let $k$ be a perfect field and $G$ a smooth and connected algebraic $k$-group. Then there exists a smallest connected linear subgroup $L$ of $G$ such that the quotient $G/L$ is an abelian variety. Furthermore, $L$ is smooth and of formation compatible with extension of $k$.

Definition: We write $av(G)$ for $G/L$ as in the Theorem.

Now fix a dvr $R$ of mixed characteristic $(0,p)$ with fraction field $K$ and residue field $k$. Let $A_K$ be an abelian variety over $K$. There exists an abelian variety quotient $B_K$ of $A_K$, unique up to isogeny, with the following properties:

1. $B_K$ has good reduction
2. Any abelian variety quotient $A_K\rightarrow C_K$ of $A_K$ having good reduction factors through $A_K\rightarrow B_K$.

If we impose the additional assumption that the kernel of $A_K\rightarrow B_K$ is connected (i.e. an abelian sub-variety of $A_K$), then $B_K$ is uniquely determined. We call this $B_K$ the maximal good reduction quotient of $A_K$.

The surjection $A_K\rightarrow B_K$ induces a mapping $A\rightarrow B$ on Neron models over $R$ and hence a mapping on identity components of closed fibers $A^0_k \rightarrow B_k$ which yields a homomorphism of abelian varieties $$\varphi:av(A^0_k)\rightarrow B_k.$$

Question: Is the kernel of $\varphi$ an abelian sub-variety of $av(A^0_k)$?