In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group $\mathrm{ G}$, a precise condition under which an $\mathrm{ R}$-orientation of a $\mathrm{ G}$-equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel$-$Whitney class of a "homogeneous" $\mathrm{ G}$-equivariant bundle with respect to an $\mathbb{ E}_\infty^{ \mathrm{ G}}$-ring spectrum $\mathrm{ R}$. As an application, we show that the $2$-fold direct sum of any homogeneous bundle is $\mathrm{ H}\underline{ \mathcal{ A}}_{ \mathrm{ G}}$-orientable, where $\underline{ \mathcal{ A}}_{ \mathrm{ G}}$ is the Burnside Mackey functor. We notice that $\mathrm{ H}\underline{ \mathcal{ A}}_{ \mathrm{ G}}$-orientability is equivalent to $\mathrm{ H}\underline{ \mathbb{ Z}}$-orientability when the order of $\mathrm{ G}$ is odd. When the order of $\mathrm{ G}$ is even, we show that a $\mathrm{ G}$-equivariant analog of the tautological line bundle over $\mathbb{ RP}^\infty$ is $\mathrm{ H}\underline{ \mathbb{ Z}}$-orientable but not $\mathrm{ H}\underline{ \mathcal{ A}}_{ \mathrm{ G}}$-orientable.
Categories: math.AT