According to the  answer of  Sebastan  and previous  edit of  Ben  McKay I revise my post as follows:


Assume that $E$ is a vector bundle over a manifold  $M$  with a connection $\nabla$.
Is there a (unique) connection
$\nabla'$ on $E':=Hom (E,E)$ with the following property;


For every curve $\gamma$ which connects point $x$ to $y$, with $\nabla$ parallel transport $\phi$ and $\nabla'$ parallel transport $\psi$, we have
$\psi(T)=\phi T\phi^{-1}$.
Moreover if $\nabla$ is a Riemannian connection corresponding to a Riemannian metric on $E$, can we choose a Riemannian comnnection $\nabla '$ as above.In the latter we consider the natural Riemannian metric inducing by initial metric on $E$ defined by $tr(AB^{*})$ on the home bundle $Hom(E,E)$.