Linear Instability of Sasaki Einstein and nearly parallel ${\rm G}_2$ manifolds

In this article we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${\rm G}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly we prove that nearly parallel $\rm G_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space ${\rm SO}(5)/{\rm SO}(3)_{irr}$ which is a $7$-dimensional homology sphere with a proper nearly parallel ${\rm G}_2$ structure.