Given a dataset $X = \{x_1, x_2, \dots, x_n\}$ with mean $\mu$ and population variance $\sigma^2$, define a new dataset $Y$ where each observation is transformed by $y_i = \theta \cdot x_i^2$. If the distribution of $X$ is symmetric about $\mu$ and has a kurtosis $\beta_2 = \frac{\mu_4}{\sigma^4}$, what is the variance $\sigma_y^2$ in terms of $\theta, \sigma^2, \mu, \text{ and } \beta_2$?