I have a 6-sheeted covering $\widetilde{X_1}$ of $X:=S^1\vee S^1$. Is this covering regular/normal? Why? Here, the rotation by 180° around some $\tilde{x_i}$ for $i\in\{1,2,3,4,5,6\}$ represents a covering transformation for $\widetilde{X_1}$. But, I think it doesn't imply that for any $i,j\in\{1,2,3,4,5,6\}$ there exists a covering transformation sending $\tilde{x_i}$ to $\tilde{x_j}.$ Also, it seems difficult to study the image of the induced homomorphism in $\pi(X,x)$. Thanks!
Using the presentation $\pi_1(S^1\vee S^1)=\langle a,b\rangle$ with $a$ and $b$ corresponding to your edge labelings, there is a surjective homomorphism $f:\pi_1(S^1\vee S^1)\to C_6$, where $C_6=\langle x\mid x^6=1\rangle$ is the cyclic group of order $6$, given by $a\mapsto x$ and $b\mapsto x$. The $6$-sheeted covering space you draw is the one associated to $\ker(f)$. Since kernels are normal subgroups, the cover is normal.
The covering transformations that D. Thomine mentions are from the action of $x$ on the cover, which shifts each $\widetilde{x}_i$ to $\widetilde{x}_{i+1}$ (with indices modulo $6$).