Let $(X,A),(Y,B)$ be topological paris and $f,g:(X,A)\rightarrow (Y,B)$ be continuous functions. (i.e. $f(A)\subset B$ and $g(A)\subset B$)

Assume that $f,g$ are homotopic so that there exists a continuous function $F:X\times I \rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$. Then, does there exist a continuous function $G:X\times I \rightarrow Y$ such that $G(x,0)=f(x)$ and $G(x,1)=g(x)$ and $G(A\times I)\subset B$?

What would be a terminology to call such homotopy $G$ and the relation of $f,g$ that admits the existence of such $G$? $G$ is definitely not "a homotopy relative to $A$". Is there some good terminology to call this?