The first contribution of this paper is the extension of the non-iterative transformation method, proposed by T\”opfer more than a century ago and defined for the numerical solution of the Blasius problem, to a Blasius problem with extended boundary conditions. This method, which makes use of the invariance of two physical parameters with respect to an extended scaling group of point transformations, allows us to solve numerically the Blasius problem with extended boundary conditions by solving a related initial value problem and then rescaling the obtained numerical solution. Therefore, our method is an initial value method. However, in this way, we cannot fix in advance the values of the physical parameters, and if we need just to compute the numerical solution for given values of the two parameters we have to define an iterative extension of the transformation method, which is the second contribution of this work.