Goals per match with a Gamma prior

You model the number of goals a team scores per match as a Poisson process with unknown average rate $\lambda$ (goals per match). Your prior is $\text{Gamma}(\alpha=2, \beta=2)$, with shape $\alpha$ and rate $\beta$. Across 4 matches the team scores a total of 6 goals.

Using Bayesian updating, what is the posterior mean estimate of the scoring rate $\lambda$?