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Vehicular networks have been developed to enhance transportation system safety, security, and efficiency. It enables new mobile applications and services like Intelligent Transport Systems (ITS) safety applications. As these applications usually are delay-sensitive, a fast and secure handoff schemes for such networks should be developed. The IEEE 802.11 p Wireless Access in the Vehicular Environment (WAVE) defines amendments to IEEE 802.11 i to address important issues such as frequent disconnection and handoff so that vehicles can be ready for communications as quickly as possible. For this purpose, 802.11 p does not require any authentication process during the handoff, which causes security problems. Recently, a group-based lightweight authentication scheme to enhance the security of 802.11 p vehicular networks during handoffs is proposed. The access points are divided into different trust groups and the group session key is introduced to generate the pairwise transient key which is used by the vehicle to authenticate itself to the access points in the same group. The scheme is claimed to be secure and can resist various attacks. However, after analysis, it was found that the scheme does not preserve the privacy and anonymity of vehicles. It also does not provide backward and forward security, mutual authentication and is subject to denial of service attack. In this paper the deficiencies of the original scheme is demonstrated then the proposed improved scheme that eliminates these deficiencies is presented. These improvements introduce slight increase in computation and communication overhead.