To solve this, let's carefully analyze the symmetry of both the regular hexagon (FGHIJK) and the square (ABCD) on the coordinate plane.

Key Points:

• Hexagon Symmetry: A regular hexagon has 6 lines of symmetry, each of which either passes through opposite vertices or through the midpoints of opposite sides (these are the perpendicular bisectors).
• Square Symmetry: A square has 4 lines of symmetry: two along its diagonals and two perpendicular bisectors that go through the midpoints of its sides.

Given that:

• The hexagon and the square share the same center.
• One side of the square ($\overline{AB}$) is parallel to one side of the hexagon ($\overline{FG}$).

Analysis of Options:

1. Option A: "Any of the perpendicular bisectors of the sides of the hexagon"
   This would only hold true if the square had the same rotational symmetry as the hexagon, which it does not. Since the hexagon has 6-fold symmetry and the square only has 4-fold symmetry, the reflection across the perpendicular bisectors of the hexagon’s sides would not map the square onto itself.
   Option A is incorrect.

2. Option B: "Either diagonal of the square"
   Reflecting across a diagonal of the square would map the square onto itself, but it would not map the hexagon onto itself since the diagonals of the square do not align with the hexagon’s axes of symmetry.
   Option B is incorrect.

3. Option C: "Either perpendicular bisector of the sides of the square"
   Reflecting across the perpendicular bisectors of the square’s sides would map the square onto itself. Since one side of the square is parallel to one side of the hexagon, this reflection will also map the hexagon onto itself (as it coincides with one of the hexagon's axes of symmetry). Thus, the combined figure can reflect onto itself across the perpendicular bisectors of the square’s sides.
   Option C is correct.

4. Option D: "There are no lines across which this figure can reflect onto itself"
   Since we’ve established that the combined figure can reflect onto itself across the perpendicular bisectors of the square, this option is incorrect.
   Option D is incorrect.

Conclusion:

The correct answer is C. Either perpendicular bisector of the sides of the square.

Would you like further details on symmetry or additional questions? Here are some related questions:

1. How many lines of symmetry does a regular hexagon have?
2. How many lines of symmetry does a square have?
3. What are the properties of reflectional symmetry in polygons?
4. Can a figure with different types of symmetry share common lines of reflection?
5. How does rotational symmetry differ from reflectional symmetry?

Tip: In problems involving symmetry, always consider both the figure’s lines of reflection and its rotational symmetry!