This can be done via the UI.. it's relatively destructive, but here goes..
Enable the shipped add-ons Add Mesh : Extra Objects, and Bool
Set your Pivot to 'Individual Origins', and your Transform
Orientation to 'Normal'
Add a Mesh > Math Function > Regular Solid > Dodecahedron
ShiftG Select Similar, by Length, unwanted edges, and CtrlX ...
TBH not totally sure I'm on track here, re symmetry of desired result, will go thru the process of adding as dupliface objects, aligned to the face normals
Enable Add Mesh Extras Addon and add the dodecahedron from the presets Add Mesh > Math Functions > Regular Solids
Add a dodecaheron, dissolve the long edges crossing the pentagon
EDIT: so I've just realized that all your cones are supposed to converge towards the same point, in that case, as Robin says, you can just press E to extrude then press Enter right away, and scale down with S0. When all your cone tips are on the same point, merge by distance with AltM
A simple Extrude will extrude along the Z local axis of the selected faces....