In this paper, we develop a unified framework for transmission design and performance analysis of secondary users (SUs) in cognitive radio networks where adaptive modulation is adopted by primary users (PUs). With the developed framework, the interference from a secondary transmitter (ST) to a PU is considered. To promise the primary's quality of service (QoS), the ST's transmit power is allocated based on the gap between the effective signal-to-noise power ratio (SNR) received at the PU and the lower SNR boundary of the modulation mode that is being used by the PU. As such, the achievable data rate over the secondary link is quantified with closed-form expressions by taking into account different capabilities of the ST. Moreover, we investigate two scenarios for the primary's implementation of adaptive modulation: the scheme with feedback of the channel quality and the scheme with modulation mode feedback. For the latter, the ST has to suppose a value as the primary's channel quality, and the PU's service outage probability is defined. Numerical results substantiate the validity of the developed framework and the performance analysis conducted. Interestingly, our results indicate that the higher the peak transmit power an ST can offer, the better performance the overall network can achieve, not only for SUs but for PUs as well.