Inner Automorphisms of a Lie Subalgebra

Question

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}\subseteq\mathfrak{g}$ a subalgebra. If $\tau$ is an inner automorphism of $\mathfrak{h}$, can we always lift $\tau$ to an inner automorphism of $\mathfrak{g}$? If not, what additional hypotheses are needed?