= Solution
The composite $L_1L_2$ is the smallest subfield of $M$ containing both $L_1$ and $L_2$. Since $L_1/K$ is separable and normal, its minimal <polynomials> remain separable and split after base change to $L_2$. Hence $L_1L_2/L_2$ is Galois.
Restriction defines
$$
\operatorname{Gal}(L_1L_2/L_2)\longrightarrow\operatorname{Gal}(L_1/K),
\qquad \sigma\longmapsto\sigma|_{L_1}.
$$
It is a homomorphism because every element of the source fixes $L_2$ and therefore $K$. Its kernel fixes both generating fields $L_1$ and $L_2$, hence fixes their composite; the map is injective.
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