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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 $Hom(E,E)$.

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 $E'$- bundle

Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$. Is there a (unique) connection $\nabla'$ on $Hom (E,E)$ 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 $\psi(T)=\phi^{-1}T\phi$. 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 $Hom(E,E)$.

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 $Hom(E,E)$.

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 $\psi(T)=\phi^{-1}T\phi$. 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 Riemannian metric $tr(AB^{*})$ on the $E'$ bundle.

Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$. Is there a (unique) connection $\nabla'$ on $Hom (E,E)$ 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 $\psi(T)=\phi^{-1}T\phi$. 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 $Hom(E,E)$.