Let $G$ be a finite group of order $4p$ where $p\ge 5$ is any odd prime number and $\sigma,\tau\in G$ with $\text{ord }\sigma =2p$, $\text{ord }\tau=2$ and $\sigma^\tau :=\tau\sigma \tau ^{-1}=\sigma^{-1}$. Is $$G=\langle\sigma,\tau : \sigma^{2p}=\tau^2=1,\sigma^\tau =\sigma^{-1}\rangle$$ a valid group presentation?
In other words: Can we conclude from the given facts that $G$ has such a presentation?
By the given condition, we at least know that $G$ is a quotient of th epresented group. Remains to show that the presented group has no more than $4p$ elements. To do so you might show (by induction) that all elements of the presented group can be written as $\sigma^i\tau^j$ with $0\le i<2p$, $0\le j<2$.