This is a theorem of Bernhard Neumann and his wife (and, as he once told me, can be found in his Collection of Works; he even told me which volume I should look at, but I do not remember; I am sure mathsci can help). A "modern" proof of that fact is easy. Basically suppose that  two finite groups $G$ and $H$ are isomorphic.  Then take $F=G\times H$ and the HNN extension $T$ of $F$ with associated subgroups $G, H$. In $T$, the subgroups $G$ and $H$ are conjugate. Now, $T$ is infinite, but as every HNN extension of a finite group, $T$  is residually finite because it is virtually free (say, by Stallings ends theorem). Hence $T$ has a finite quotient $R$ such that the natural maps from $G$ and $H$ into $R$ are embeddings. Thus $G$ and $H$ are conjugated subgroups of a finite group $R$.   

**Edit:** The paper is this: MR0064042 
Neumann, B. H.; Neumann, Hanna "Partial endomorphisms of finite groups." J. London Math. Soc. 29, (1954). 434–440. 

If $G$ and $H$ are given as subgroups in the same finite group $F$, then there is no need to take $F= G\times H$.