Stiefel--Whitney classes of associated bundles

Let $Ad: H \to GL(\mathfrak{g}/\mathfrak{h})$ be the adjoint representation. Basically this this could answer your question. To be more precise: The principal bundle $G \to G/H$ has a classifying map $$ f : G/H \to BH$$ Composing this map with $BAd:BH \to BGL(k)$ will give the associated $GL(k)$ principal bundle. For these objects Stiefel-Whitney classes are defined in $H^*(BGL(k);\mathbb{Z}/2)$ and we have $w_i(G \to G/H) = f^* BAd^* w_i.$ Explicelty the first Stiefel-Whitney class is induced by the determinant $$ B\det: BGL(k) \to B\mathbb{Z}/2.$$ Since $Bdet \circ BAd = B(det\circ Ad)$ and $det \circ Ad \equiv const$ this gives $$w_1(G\to G/H) = f^* \underbrace{BAd^*Bdet^*}_{=0}x = 0$$ where $x \in H^1(B\mathbb{Z}/2,\mathbb{Z}/2) $ is the generator. But the whole point is that the adjoint representation takes values in orientation preserving automorphisms.