Extending field homomorphisms in the general setting

Question

If we have arbitrary field extensions $\tilde{K}/K$, $\tilde{L}/L$ and a homomorphism $\sigma:K\rightarrow L$. Under what conditions there exists a homomorphism $\tilde{\sigma}:\tilde{K}\rightarrow \tilde{L}$ that extends $\sigma$? and if such homomorphism exist, is there a way to count the number of possible $\tilde{\sigma}$ in function of the separability degree/degree of the extensions (possibly $[\tilde{L}/\sigma(K)]_s$)?