I have met the following problem. A group $G$ is given as follows

$G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$

Is the subgroup generated by $y$ and $t$ just the group $\langle y,t|t^{-1}yt =y^2 \rangle$? It seems not right, because the first relation and the third relation may be able to give more relations on $y$ and $t$. But I could not prove it.

And the general question would be given a group by a presentation, is there any method to determine the subgroup generated by a subset of the generators?

Sorry, I try to edit the question, but deleted it, so I have to post it again.