Lifting surjective von Neumann algebra homomorphisms

Yes you can get a $phi$ that is a homomorphism. Here is a quick sketch.

First let $p=sup$ {$p_alpha,$ projections in $Ker theta$}. So $pin Ker theta$. Furthermore $pin Z(M)$, the center of M.

To see this note that if this were not true then we could find a unitary $uin M$ with $pneq upu^star$. So then $pwedge upu^star$ would be a projection in $Ker theta$ bigger than $p$.

From here you can get that $Ker theta=pMp$, and so we can decompose $M=Ker theta oplus M_1$ and $theta|_{M_1}$ is injective and thus an isomorphism, thus $phi$ can be chosen to just be the inverse of $theta$ on $M_1$.

Note that if we demand that $phi$ be unital, this doesn't work and I don't think it can be done in general. I will have to give it more thought.