Group cohomology of elementary abelian group of prime-fourth order

Suppose $$p$$ is a prime number. We are interested in the elementary abelian group of prime-fourth order

$$E_{p^4} = (\mathbb{Z}/p\mathbb{Z})^4 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$$

Over the integers
The homology groups over the integers are given as follows:

$$\! H_q(E_{p^4};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{\frac{2q^3 + 15q^2 + 34q + 45}{24}}, & \qquad q = 1,3,5,\dots \\ (\mathbb{Z}/p\mathbb{Z})^{\frac{2q^3 + 15q^2 + 34q}{24}}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.$$

The first few homology groups are given as follows: