This is exercise 3 from Munkres section 84.
Let $X$ be the wedge of two circles; let $p:E\rightarrow X$ be a covering map. The fundamental group of $E$ maps isomorphically under $p_\ast$ onto a subgroup $H$ of the fundamental group of $X$; the latter is free on two generators $\alpha$ and $\beta.$
(1) For each of the four covering spaces $E$ given in Exercise 2 of $\S 81$, determine the cardinality of a system of free generators for the fundamental group of $E$.
(2) For each of these covering spaces, find, in terms of $\alpha$ and $\beta$, a system of free generators for the subgroup $H$ of the fundamental group of $X$.
Here is one of $E$ given in the exercise:
There are three more covering spaces but I will try to apply to other covering spaces after I learn how to do this one. Any help is greatly appreciated. Or any reference to similar questions is welcome.