Uniform approximation by indestructible Blaschke products

We address the question: Are the inner functions in the uniform closure in H∞ of the indestructible Blaschke products? We show, in particular, that every inner function with countable spectrum is in the closure of the indestructible Blaschke products, that every Blaschke product is a product of two indestructible Blaschke products and we study approximation in modulus.