To solve the problem, let's go step by step:

1. **Identify Set \( B \):**

The set \( B \) is given by:

\[
B = \{ x \in \mathbb{Z} \, | \, |x| \leq 4 \}
\]

Therefore, \( B = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\} \).

2. **Determine \( X \):**

We need to find a set \( X \) such that:

- \( A \subseteq X \subseteq B \)
- \( A \cup X = B \)
- \( X \) has exactly 4 elements.

3. **Analyze Conditions:**

Since \( A \subseteq X \), \( X \) must include all elements in \( A \). So \( X \) must at least contain:

\[
\{-4, -2, -1, 2, 3, 4\}
\]

Since \( X \) has 4 elements, find 4 elements from \( A \) to make \( A \cup X = B \).

4. **Ensure \( A \cup X = B \):**

To ensure \( A \cup X = B \), \( X \) must include the missing elements from \( B \) that are not in \( A \) and without exceeding 4 elements in total.

- Missing elements from \( B \) are \( \{-3, 0, 1\} \).
- Choose elements from \( A \) to replace and add some of the missing elements.

5. **Construct \( X \):**

- From \( A = \{-4, -2, -1, 2, 3, 4\}\), select elements ensuring the inclusion of missing \( B \) elements and making \( X \) have 4 elements.
- Selecting \( X = \{-3, 0, 1, 4\} \).

This \( X \) satisfies:

- \( A \subseteq X \subseteq B \)
- \( A \cup X = B \)
- \( X \) has 4 elements.

Thus, the set \( X \) is \(\{-3, 0, 1, 4\}\).