tensor products of the corresponding maps gives an inclusion of $H$–modules from $M:=M_\chi \otimes \Bbbk \chi^{-1} \to (N \otimes
  \Bbbk[G])|_H$. As the image of $M$ inside of $N \otimes \Bbbk[G]$ lies in $N \otimes \Bbbk f$, that is finite dimensional rational $G$–module, the proof of the first assertion is finished. The second assertion follows directly from the first. It only remains to prove that if $\chi$ is an arbitrary character of an observable $H$, then $\chi$ is exendible. Given $\chi$, an arbitrary character of $H$, we can find a finite dimensional $G$–module $N$ and an $H$–inclusion of $\Bbbk_\chi \to N$. If we call $n \in N$ the image of $\chi$, we have that for all $x \in H\,,\, x\cdot n= \chi(x)n$. If $\alpha \in N^\vee$ is a linear functional such that $\alpha(n)=1$ and take the polynomial $\alpha|n \in \Bbbk[G]$ (recall that $(\alpha|n)(y)=\alpha(y\cdot n)$ for all $y \in G$). It is clear that if $x \in H$ then $(x \cdot
  (\alpha|n))(y)=(\alpha|n)(yx)=\alpha((yx)\cdot n)= \alpha(y\cdot (x \cdot n))=\alpha(y
  \cdot (\chi(x)n))=\chi(x)\alpha(y \cdot n)= \chi(x) (\alpha|n)(y)$. Moreover, $(\alpha|n)(1)=\alpha(n)=1$.

It seems that the main motivation of the authors of [@kn:homolie; @kn:mostow] to study the extension of representations from normal Lie subgroups to the whole group, was 