For a given closed oriented smooth 4-manifold $M$, fix a metric on it, we can have the orthonormal frame bundle of its tangent bundle to be a principle $SO(4)$-bundle. This bundle can be lifted to a principle $Spin^{\mathbb{C}}$-bundle, denote it as $P$.This gives a $Spin^{\mathbb{C}}$-structure over $M$.
I have a few questions:
How many $Spin^{\mathbb{C}}$-structures do we have (up to isomorphism of principal bundles without changing its covering over the frame bundle)? Countably many or uncountably many?
How strict is the $Spin^{\mathbb{C}}$-structure depending on the metric chosen? e.g. if we have s $Spin^{\mathbb{C}}$-structure exists for a metric, does that ensure the $Spin^{\mathbb{C}}$-structure exists for all metric?
What does the set of all $Spin^{\mathbb{C}}$-structures when fixing $g$ look like? I did encounter in some text saying that it is isomorphic to $H^2(X;\mathbb{Z})$, but I don't see why.
Any comment is appreciated.