Need help creating a pentagon network

I think this might be what you seek:

Graphics[{
    FaceForm[GrayLevel[0.5]], EdgeForm[{Black, Thick}],
    RegularPolygon[5],
    FaceForm[GrayLevel[0.8]],
    RegularPolygon[#, {1, 3 Pi/10}, 5] & /@ CirclePoints[{2 Cos[Pi/5], 3 Pi/10}, 5],
  }
]

All pentagons have radius 1, for convenience. The central pentagon is centered on the origin, so the outer pentagons have centers at a distance equal to twice the apothem of the central pentagon, and at positions equally spaced around a circle which can be obtained from CirclePoints using an appropriate angular shift.

Looking at the figure, I noticed that the outer pentagons all have the same angular with respect to the horizontal axis, which simplifies matters considerably, so they are constructed using RegularPolygon using the center locations obtained from CirclePoints above, and with a constant radius and angular position with respect to the $x$ axis.

PolyhedronData["Dodecahedron", "Net"] /. Polygon[x_] :> Polygon[Take[x, 6]]