If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$,
$\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring spectrum
$\mathit{KO}$. Using complex vector bundles, we get another generalized cohomology theory, represented by
$\mathit{KU}$. Using quaternionic vector bundles, we get a third one, represented by $\mathit{KSp} \simeq
\Sigma^4\mathit{KO}$.

In the same way, one can define the Grothendieck ring of oriented vector bundles $\mathit{KSO}(X)$, spin vector
bundles $\mathit{KSpin}(X)$, and so on for any [$G$-structure](https://ncatlab.org/nlab/show/G-structure). Do these
constructions extend to spectra $\mathit{KSO}$, $\mathit{KSpin}$, and so on? If not, what issues arise?

The first concern I would imagine is a failure of Bott periodicity, but $\pi\_k(\mathit{SO})$ and
$\pi\_k(\mathit{Spin})$ agree with $\pi\_k(O)$ for $k\ge 2$, so in at least these cases some sort of
construction might be possible.

An alternative approach sidestepping Bott periodicity would be to use algebraic $K$-theory: if $\mathsf{Vect}\_{k}$ denotes the topological
symmetric monoidal category of $k$-vector spaces, then $K(\mathsf{Vect}\_{\mathbb R}) = \mathit{ko}$ and
$K(\mathsf{Vect}\_{\mathbb C}) = \mathit{ku}$. So if $\mathsf{Vect}^{\mathrm{or}}$ denotes the topological symmetric monoidal
category of *oriented* real vector spaces, would it be reasonable to define $\mathit{kso}:= K(\mathsf{Vect}^{\mathrm{or}})$? Is
this an interesting object? This could also generalize to $G$-structures.

Someone must have thought about this, but I can't find it written down anywhere.

**Edit**: Though I've accepted Denis' excellent answer, I would still be interested to learn of existing references which consider these kinds of cohomology theories/spectra.