construct an unbiased estimator for variance based on X1 and X2

Question

I have a couple of questions: Suppose $X_{1},...X_{n} \stackrel{iid}{\sim} N(0,\sigma^2)$, where $\sigma^2$ $\in R^{+}$ is unknown. I need to find a sufficient statistic for $\sigma^2$, an unbiased estimator of $\sigma^2$ based on $X_{1}$ and based on $X_{1},X_{2}$, For the sufficient statistic, I have $\sum_{i=0}^n X_{i}^2$ The estimate based on $X_{1}$ is $X_{1}^2$ and the estimate based on $X_{1},X_{2}$ is $\frac{\\X_{1}^2+X_{2}^2}{2}$. Now, when I tried to compute the conditional expectation of the first estimate given the sufficient statistic and the conditional expectation of the second estimate given the sufficient statistic, I have the same result $\frac{sum_{i=0}^n X_{i}^2}{n}$. The next question asks to calculate the variance of the two conditional expectations. That means one of my expectations is probably incorrect. I'm wondering if the unbiased estimator based on two data points is also correct...Any help will be appreciated.