Let us say that an algebra $A$ over a field $k$ is Picard-surjective if the canonical map $$ \mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$ is surjective. Here $\mathrm{Pic}(A)$ denotes the group of isomorphism classes of invertible $A$-$A$-bimodules and the map sends an automorphism $\alpha$ to the $A$-$A$-bimodule $A_\alpha$, where the left action is the usual one and the right action is via $\alpha$.

**Q:** For any given finite-dimensional $k$-algebra $A$, does there exist a Morita-equivalent one that is Picard-surjective?

If not, I am interested in conditions under which this is true. I am mainly interested in the case $k=\mathbb{R}$ or $\mathbb{C}$, and for all the examples that I came up with so far, this seems to be correct, as far as I can tell.