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I have a question about the fundamental group of the covering space of $S^1 \vee S^1$ from part (6) of the following table from Hatcher's book enter image description here

Hatcher claims that all the fundamental groups from the table can be computed using Van Kampen's theorem, but thinking a few hours about the space from (6), I still can not find the right covering to compute the fundamental group using Van Kampen's theorem. Any hint/solution will be highly appreciated.

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    $\begingroup$ One is not restricted to a single application of Van Kampen's Theorem. Applying it multiple times is a perfectly valid method. $\endgroup$
    – Lee Mosher
    Nov 20 at 23:56

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