To solve this problem, let’s analyze the given information carefully and apply it step-by-step:

1. Matrix Properties and Given Equations
   We are given a $2 \times 2$ matrix $B$ with integer entries such that: $B^3 = 4B$ and $\det(B) = -12.$ We need to determine the distinct values the trace of $B$ can take modulo 5.

2. Eigenvalues Analysis The condition $B^3 = 4B$ implies a relationship between the eigenvalues of $B$. Suppose $\lambda$ is an eigenvalue of $B$. Then $\lambda$ must satisfy: $\lambda^3 = 4\lambda.$ This equation can be factored as: $\lambda(\lambda^2 - 4) = 0,$ or $\lambda(\lambda - 2)(\lambda + 2) = 0.$ Therefore, the possible eigenvalues of $B$ are $\lambda = 0$, $\lambda = 2$, or $\lambda = -2$.

3. Trace and Determinant Constraints For a $2 \times 2$ matrix $B$ with eigenvalues $\lambda_1$ and $\lambda_2$:

   • The trace of $B$ is $\text{tr}(B) = \lambda_1 + \lambda_2$.
   • The determinant of $B$ is $\det(B) = \lambda_1 \lambda_2$.

   Given that $\det(B) = -12$, we can write: $\lambda_1 \lambda_2 = -12.$ We now examine the pairs $(\lambda_1, \lambda_2)$ that satisfy both $\lambda_1 \lambda_2 = -12$ and $\lambda_1, \lambda_2 \in \{0, 2, -2\}$.

4. Finding Compatible Eigenvalues Since none of the products $0 \cdot 0$, $2 \cdot 2$, $(-2) \cdot (-2)$, or $2 \cdot (-2)$ equal $-12$, this suggests that we need to consider values of $\lambda_1$ and $\lambda_2$ beyond $\{0, 2, -2\}$. The integer solutions must also consider larger values like $\lambda_1 = 3$ and $\lambda_2 = -4$ (or vice versa).