I'm currently stuck at a mathematical problem and I really don't know where to start, since I'm not an expert in algebra over finite fields.
Define an $\mathbb{F}_{5}$-linear map $\varphi$ on $\operatorname{Map}(\mathbb{F}_{5}, \mathbb{F}_{5})$ such that $\operatorname{Im}(\varphi)$ = $\operatorname{Map}_{even}(\mathbb{F}_{5} , \mathbb{F}_{5})$.
For a map $f:\ \Bbb{F}^5\ \longrightarrow\ \Bbb{F}^5$ to be even means that $f(-x)=f(x)$ for all $x\in\Bbb{F}^5$. In other words, replacing $x$ by $-x$ in this function leaves its value unchanged. For example, the functions $$f(x)=x^2\qquad\text{ or }\qquad f(x)=g(x)g(-x),$$ work, where $g:\ \Bbb{F}^5\ \longrightarrow\ \Bbb{F}^5$ can be any function. Can you think of more such functions?
Now to construct such a morphism $\varphi$, you need to construct some $\varphi(f)\in\operatorname{Map}_{\operatorname{even}}(\Bbb{F}^5,\Bbb{F}^5)$ from every $f\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$, in such a way that $$\varphi(\lambda f)=\lambda\varphi(f)\qquad\text{ and }\qquad\varphi(f+g)=\varphi(f)+\varphi(g),$$ holds for all $\lambda\in\Bbb{F}^5$ and all $f,g\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$. And moreover, you want $\varphi$ to be surjective, so every even function should be constructible from some $f\in\operatorname{Map}(\Bbb{F}^5,\Bbb{F}^5)$ in this way.
The surjectivity is not something to worry about at first; once you see what kinds of constructions work to get even functions, it should not be hard to find one that makes $\varphi$ surjective. So my advice is to go wrestle with how you can construct even functions from general functions.