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Butter a 25 x 32cm swiss roll tin and line with parchment. Whisk the eggs and sugar in a stand mixer until thick, about 5 mins. Fold in the flour, coconut and lime zest. Spread into the tin. Bake for 14-16 mins until golden. Invert onto a large sheet of parchment dusted with 1 tbsp sugar. While still warm, roll the cake up into a spiral along one of the short sides, with the parchment inside. Leave to cool fully.
For the buttercream, beat the butter in a stand mixer until pale, then the icing sugar in three parts, followed by the lime juice. Unroll the cooled sponge and spread with 100g of the buttercream before rolling back up. Put on a board. Divide the rest of the buttercream between two bowls. Colour one dark green and the other light. Spoon into two piping bags fitted with a multiple opening nozzle. Pipe the green buttercreams in alternating stripes along the length of the cake.
Lay the strawberry laces between the stripes. Melt the chocolate in a heatproof bowl set over a pan of simmering water. Spoon an 8cm circle of melted chocolate onto a sheet of parchment. Chill until set, about 10 mins. Make the face using liquorice allsorts and strawberry laces, and attach to the chocolate disc using more chocolate. Attach to one end of the cake using buttercream. Stick liquorice laces into the top and sides to make antennae and legs. Will keep chilled for up to three days.
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