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An orbifold datum is a collection $\mathbb{A}$ of algebraic data in a modularfusion category $\mathcal{C}$. It allows one to define a new modular fusioncategory $\mathcal{C}_{\mathbb{A}}$ in a construction that is a generalisationof taking the Drinfeld centre of a fusion category. Under certain simplifyingassumptions we characterise orbifold data $\mathbb{A}$ in terms of scalarssatisfying polynomial equations and give an explicit expression which computesthe number of isomorphism classes of simple objects in$\mathcal{C}_{\mathbb{A}}$.  In Ising-type modular categories we find new examples of orbifold data which- in an appropriate sense - exhibit Fibonacci fusion rules. The correspondingorbifold modular categories have 11 simple objects, and for a certain choice ofparameters one obtains the modular category for $sl(2)$ at level 10. Thisconstruction inverts the extension of the latter category by the $E_6$commutative algebra.

Fibonacci-type orbifold data in Ising modular categories