Preheat the oven to 180C (160C fan-assisted). Put the softened butter, oil and caster sugar in a large mixing bowl. Beat with an electric mixer or by hand until creamy. Crack the eggs into the bowl and add the Baileys. Beat again until just combined.
Sieve in the flour and cocoa powder and fold everything together. It's fine if it's a bit lumpy.
Line a 12-hole muffin tray with muffin cases. Divide the batter evenly between them, making sure to level off so that the cupcakes rise evenly.
Bake for about 20 minutes or until risen and firm. A skewer pushed into the centre of a cupcake towards the middle of the tray should come out clean.
Transfer your cupcakes to a wire rack and leave them to cool completely.
Make the Baileys buttercream icing
Place the butter in a large bowl. Whisk until soft.
Sift in the icing sugar and cocoa powder.
Add the Baileys, then whisk starting slowly until just combined, then whisk on high until the frosting is pale, creamy and significantly increased in volume.
Transfer the buttercream icing into a piping bag fitted with a star nozzle and pipe on top of the cupcakes.