Gluing to diffeomorphisms to get a diffeomorphism

Question

Let $M,N_1,N_2$ be smooth manifolds with boundary of the same dimension. Suppose $f_i:\partial M\to \partial N_i$ are diffeomorphisms for $i=1,2$, and $F:N_1\to N_2$ is also a diffeomorphism. Then $g:=f_2^{-1}\circ F\circ f_1$ is a self-diffeomorphism of $\partial M$. Suppose $g$ extends to a diffeomorphism $G$ of $M$. Then we can glue $F$ and $G$ to get a well-defined homeomorphism $H:M\cup_{f_1}N_1\to M\cup_{f_2}N_2$. Is it true that $H$ is a diffeomorphism? (I cannot show smoothness of $H$ at the common boundary.) Or can we deform $F$ and $G$ to make $H$ a diffeomorphism?