  For a given function $F$ from $\mathbb F_{p^n}$ to itself, determining
whether there exists a function which is CCZ-equivalent but EA-inequivalent to
$F$ is a very important and interesting problem. For example, K\"olsch
\cite{KOL21} showed that there is no function which is CCZ-equivalent but
EA-inequivalent to the inverse function. On the other hand, for the cases of
Gold function $F(x)=x^{2^i+1}$ and $F(x)=x^3+{\rm Tr}(x^9)$ over $\mathbb
F_{2^n}$, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and
Leander) \cite{BCP06, BCL09FFTA} found functions which are CCZ-equivalent but
EA-inequivalent to $F$. In this paper, when a given function $F$ has a
component function which has a linear structure, we present functions which are
CCZ-equivalent to $F$, and if suitable conditions are satisfied, the
constructed functions are shown to be EA-inequivalent to $F$. As a consequence,
for every quadratic function $F$ on $\mathbb F_{2^n}$ ($n\geq 4$) with
nonlinearity $>0$ and differential uniformity $\leq 2^{n-3}$, we explicitly
construct functions which are CCZ-equivalent but EA-inequivalent to $F$. Also
for every non-planar quadratic function on $\mathbb F_{p^n}$ $(p>2, n\geq 4)$
with $|\mathcal W_F|\leq p^{n-1}$ and differential uniformity $\leq p^{n-3}$,
we explicitly construct functions which are CCZ-equivalent but EA-inequivalent
to $F$.
