Finite ${\mathbb{Z}}/2{\mathbb{Z}}$-CW complexes which are not homotopically stratified by orbit type

For $k \geq 2$, we construct finite ${\mathbb{Z}}/2{\mathbb{Z}}$-CW complexes with one ${\mathbb{Z}}/2{\mathbb{Z}}$-cell in dimensions 0, $1$ and $k+1$. Using a theorem of Bruce Hughes, we show that these complexes are not homotopically stratified by orbit type in the sense of Quinn.