The authors previously presented a randomized procedure for nonorthogonal saturated main effect fractional replicates in an $s^n$-factorial and presented an unbiased estimator of the main effect parameter vector. However, the explicit expression of the variance of the estimator remained an unsolved problem. In this paper our attention is restricted to a $2^n$-factorial, and the randomized procedure is extended to any preassigned parameters in a $2^n$-factorial system. An explicit expression of the variances of unbiased estimators of the parameters is presented. Also, in a $2^n$-factorial, some invariant properties of the information matrices and variances of the estimators in the randomized fractional replicates and a semi-invariant property of alias schemes of the fractional replicates are obtained.