Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is a canonical $n-k$ bundle $E$ over $M$. $E$ is the kernel bundle:$$\{(A,v)\in M_{n}(\mathbb{R})\times \mathbb{R}^{n}\mid Av=0\}$$
We consider the $Z_{2}$ action on base space $M$ via transpose operation
Question: Can one say that: This action can not be lifted to a $\mathbb{Z}_{2}$- equivariant structure for $E\to M$?