Fundamental group of unit sphere with three diameters

I want to compute the fundamental group of a unit sphere $S^2$ placed at origin with three diameters along the three axes in $\mathbb{R}^3$. I understand that a sphere with one diameter is equivalent to the wedge sum of a sphere and a circle and thus, its fundamental group is $\mathbb{Z}$. But what happens in the case of more than one diameter? Also, I want to compute the fundamental group based at point $(1,0,0)$. Does the base point matter for the fundamental group of this space or it would be same for any point in my space? Can someone give me any hint on how to proceed?

Thanks

But what happens in the case of more than one diameter?

A sphere with two diameters is homotopy equivalent to the wedge sum of a sphere with a bouquet (wedge sum) of three circles. To see this:

As shown in the sketch below, you now have $S^2\vee S^1\vee S^1\vee S^1$. Hence the fundamental group is free group on three generators.
Also, I want to compute the fundamental group based at point $(1,0,0)$. Does the base point matter for the fundamental group of this space or it would be same for any point in my space?