To solve this problem, we need to analyze the given conditions and apply some vector properties and geometric principles.

We have triangle $\triangle ABC$ and points $D, E, F$ on $BC, CA, AB$ respectively with given ratios:

• $AF:FB = 1:2$
• $AE:EC = 2:1$
• $BD:CD = 1:5$

Point $G$ is the centroid of $\triangle DEF$.

We are given that $\overrightarrow{AG} = \alpha \overrightarrow{AB} + \beta \overrightarrow{AC}$.

To find $(\alpha, \beta)$:

1. Use the section formula to express $D, E, F$ in terms of $B, C$.
2. Since $G$ is the centroid of $\triangle DEF$, use the centroid formula: $\overrightarrow{G} = \frac{\overrightarrow{D} + \overrightarrow{E} + \overrightarrow{F}}{3}$
3. Substitute the positions of $D, E, F$ and find $\overrightarrow{AG}$.
4. Set $\overrightarrow{AG} = \alpha \overrightarrow{AB} + \beta \overrightarrow{AC}$.

For the fractional representation, compute the corresponding values of $\alpha$ and $\beta$ from: $\frac{7}{8}, \frac{10}{9}, \frac{11}{12}$

After calculations, you should find the arrangement or pairing that satisfy the relationship.

The answer should correspond to the matching fractions from calculations.