Averaging an MVUE with an inefficient estimator

If $T_1$ is an MVUE of $\gamma(\theta)$, $\theta\in\Theta$ and $T_2$ is any other unbiased estimator of $\gamma(\theta)$ with efficiency $e<1$, then which of the following is/are correct?

I. $T=(T_1+T_2)/2$ is MVUE of $\gamma(\theta)$

II. $\operatorname{Var}(T)=\operatorname{Var}(T_1/4)$

III. $T$ cannot be MVUE for $\gamma(\theta)$

IV. $T$ is unbiased for $\gamma(\theta)$