beta binomial hierarchical model with two groups, inference on the group hyperparameters

The problem I want to solve:

Lets imagine that I have two factories A and B, where each factory produces coins. What I suspect is that the probability of tails (denoted as $\theta$) varies substantially between the two factories.

How is my data:

My observations are 10 draws for each of the three coins in factory A and another 10 draws for each of three coins in factory B

How I would model the problem: I would use a beta binomial hierarchical model with the following form:

For coin $j$ in factory A the sequence of 10 draws comes from a binomial distribution of the form $yi∼Binomial(N=10,\theta_j)$, and then $\theta_j$ values depend on a beta distribution $\theta_j∼Beta(α,β)$ (which I suspect that it is different for A and B factories).

What is my problem

On the different textbooks I have check ( Gelman et al BDA3 and Kruschke DBDA 2014) the hierarchical models are used for doing inferences on the $\theta_j$ parameter of the binomial distributon. However, in this case what I wanna check is if the parameters $\alpha$ and $\beta$ of the beta distribution is different between the two factories.

My question

Finally, as indicated above, what I want to know is if the $Beta$ distribution for factory $A$ and factory $B$ differs.