Let $F$ be the function from example 6.20 c) in [1]. That is, fix a perfect nowhere dense subset $E$ of $[0, 1]$ such that $0, 1\in E$ such that $0 < |E| < 1$, for example the fat Cantor set. Write $(0, 1)\setminus E$ as a countable union of open intervals $(0, 1)\setminus E = \bigcup_{n=1}^\infty I_n$. Let $\rho_n$ be the length of a largest subinterval of $[0, 1]$ disjoint from $I_1, ..., I_n$. Let $A = E\cup \{c_n : n\in\mathbb{N}\}$ where $c_n$ is the middle point of $I_n$. Define $G:A\to\mathbb{R}$ as $$G(x) = \begin{cases} 0, & x\in E, \\ |I_n|+\rho_n, & x = c_n\end{cases}$$ and extend $G$ to a continuous function $F:[0, 1]\to\mathbb{R}$ by letting $F$ to be linear on every open interval $J\subseteq A^c$ with end-points in $A$.
The function $F$ is $ACG$ but it has no a.e. derivative so $F$ is not in $ACG_*$. Then $F_{ap}'$ exists a.e. and is Khintchine integrable. However it cannot be Henstock-Kurzweil integrable, since supposing otherwise, if $G$ is the indefinite integral of $F_{ap}'$, then $F-G$ is in $ACG$ and $$(F-G)_{ap}' = F_{ap}'-G_{ap}' = F_{ap}'-G' = F_{ap}'-F_{ap}' = 0\text{ a.e.}$$
so from uniqueness of Khintchine integral we would have $F-G = \text{const.}$ which implies that $F$ is $ACG_*$. $\unicode{x21af}$
Also see example 16.19 in [1] for a function which is in $ACG$ but not in $ACG_\Delta \supseteq ACG_*$. The same argument shows that this function is Khintchine integrable but not Henstock-Kurzweil integrable.
- [1] "The Integrals of Lebesgue, Denjoy, Perron, and Henstock" by Russell A. Gordon
