Here is a simple example. Let $H=\mathbb C^2$ and $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$. Then $\{A,A^*\}''=M_2(\mathbb C)$, while
$$
\{A\}''=\left\{\begin{bmatrix}a&b\\0&a\end{bmatrix}:\ a,b\in\mathbb C\right\}.
$$
The papers of Zimmer are good references, but they only cover the commutative case (the case of an action on a measure space). For actions of (locally compact) groups on general von Neumann algebras, amenability is defined and studied by Claire Anantharaman-Delaroche in the following papers:
Claire Anantharaman-Delaroche,
Action moyennable d'un groupe localement compact sur une algèbre de von Neumann.(French. English summary)Math. Scand.45(1979), no.2, 289–304. https://www.mscand.dk/article/view/11844
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann. II.(French. English summary)[Amenable action of a locally compact group on a von Neumann algebra. II] Math. Scand.50(1982), no.2, 251–268.
https://www.mscand.dk/article/view/11958
In the first paper it is already proved that the crossed product $M\bar\rtimes_\alpha\Gamma$ if injective if and only if $M$ is injective and the action $\alpha$ is amenable. Here $\Gamma$ is a discrete group. For general locally compact groups only one direction of this holds.
For actions of (discrete) groups on $C^*$-algebras, amenability is defined and studied in the follow up paper:
Claire Anantharaman-Delaroche, Systèmes dynamiques non commutatifs et moyennabilité.(French)[Noncommutative dynamical systems and amenability] Math. Ann.279(1987), no.2, 297–315. https://link.springer.com/article/10.1007/BF01461725
In this paper it is proved that for an action $\alpha$ of a (discrete) group $\Gamma$ on a $C^*$-algebra $A$, the crossed product $A\rtimes_\alpha \Gamma$ is nuclear if and only if $A$ is nuclear and $\alpha$ is amenable.
Amenable actions of locally compact groups were only defined and studied recently in the preprint (still not published):
https://arxiv.org/abs/2003.03469
Amenability and weak containment for actions of locally compact groups on $C^*$-algebras, by Alcides Buss, Siegfried Echterhoff, Rufus Willett
Further references, and historical background, can be found in that preprint.
Best Answer
There are easy examples of non-extending $\ast$-homomorphisms. For instance recall that a groups homomorphism $\Phi: G \to H$ extends to the reduced $C^\ast$-algebras iff $\mathrm{ker}(\Phi)$ is amenable, but it extends to the reduced von Neumann algebras iff $\mathrm{ker}(\Phi)$ is compact. Take the trivial character of an amenable group $G$, $$ \chi \Big( \sum_{g\in G} a_g \lambda_g \Big) = \sum_{g\in G} a_g $$ always extends to a $\ast$-homomorphism $\chi:C_{\mathrm{red}}^\ast G \to \mathbb{C}$ but never pass to the von Neumann algebra unless $G$ is finite.
In the normal case: Any $\ast$-homomorphism between von Neumann algebras is spatially implemented and of the form $a \mapsto p ( a \otimes 1 ) p$, where $p$ is a projection commuting with $A \otimes \mathbb{C} 1$. Then your criterion will be something like asking the $\ast$-homomorphism to be spatially implemented by a map $V: \mathcal{H} \to \mathcal{H'}$ and take $p$ to be the range projection.
In the non-normal case: I do not have an answer. Take $A = c_0(\mathbb{N}) + \mathbb{C} 1 \subset B(\ell^2(\mathbb N))$ the $C^\ast$-algebra of sequences with a limit. There is a $\ast$-homomorphism $\psi(a_n)_n = \lim_n a_n$. It does not extends to $A^{''} = \ell^\infty(\mathbb{N})$ in a normal way but there are uncountably many non-normal extensions, one for every limit with respect to a proper ultrafilter $$ \bar\psi((a_n)_n) = \lim_{n, \, \omega} a_n $$