I constructed the non-abelian groups of order $16$ and listed the structure descriptions. The result was :
16 (C4 x C2) : C2 C4 : C4 C8 : C2 D16 QD16 Q16 C2 x D8 C2 x Q8 (C4 x C2) : C2
The group (C$4$ x C$2$) : C$2$ appears twice. Obviously, two non-isomorphic groups with this structure exist.
What are these groups and how do they differ ?
A similar result appears for order $20$ :
20 C5 : C4 C5 : C4 D20
What are the non-isomorphic groups with structure $C5:C4$ ?